AmTHMETIC 


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I9I9 

REPORTor  PROGRESS 

DULUTH  PUBLIC  SCHOOL.^ 


Y   ^    V  •  ..  :i^.~ 


This  book  is  DUE  on  the  last  date  stamped  below 


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DEC  l3  Idtf 

JUL  Jti   i    

'^'=^'0  1  4  1931 
1941 


Southern  Branch 
of  the 

University  of  California 

Los  Angeles 

Form  i,  1 

J  s  7  0 
B  ^% 

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;-"^->^- 


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VERSITY  OF  CALIFORNIA, 

LIBRARY, 

ARITHMETIC 


1919 
REPORT    OF    PROGRESS 

DULUTH  PUBLIC  SCHOOLS. 

82390 


■  V.  I 

CONTENTS. 

AIMS 1 

BRIEF  SURVEY 1 

GENERAL  DIRECTIONS 6 

Identical  Forms  for  all  Grades 6 

Checks 8 

Principles  of  Method 11 

Provision  for  Individual  Differences 13 

Standard  Practice  Material 14 

Standard  Tests  in  Arithmetic 15 

GRADE  I-B  and  I-A 18 

GRADE  II-B : 20 

GRADE  II-A 2(5 

GRADE  III-B 34 

GRADE  III-A \ . 40 

(iRADE  IV-B 47 

GRADE  IV-A 54 

(tRADE  V-B 61 

(4RADE  V-A 66 

(iRADE  VI-B 71 

GRADE  VI-A 76 

(iRADE  VII-B 82 

GRADE  VII-A 87 

GRADE  VIII-B 91 

GRADE  VIII-A 96 

GRADE  IX-B 101 

(;RADE  IX-A ! 105 

GENERAL  BIBLIOGRAPHY 108 


Will  each  teacher  i)lease  make  the  following  correctiou: 

Page  87,  line  3  should  read,  "This  will  involve  column  addition." 


PREFACE 

This  ])anii)hlet  ou  .\rithnietic  is  oue  ot"  a  series  of  five  which 
constitutes  the  Course  of  Study  so  far  available  in  jjrinted  form  tor 
use  in  the  Public  Schools  of  the  City  of  Duluth.  The  other  panijth- 
lets  are  as  follows:  one  on  ]\Iusic  and  Physical  Education;  one  on 
Geography,  History  and  Nature  Study;  one  on  Drawing-  and  In- 
dustrial Art;  one  ou  English  including  lieadiug,  Phonics,  Spelling, 
PennianshiiJ,  Language,  Composition,  Grammar,  and  Literature. 

This  Course  of  Study  was  constructed  during  the  school  term  of 
1918-1919  and  during  the  summer  of  1919.  It  was  introduced  in 
September  1919.  It  is  the  product  of  the  combined  effort  of  the 
teachers,  principals,  and  supervisors  in  the  Public  Schools  and  the 
State  Normal  School  of  Duluth. 

The  general  supervision  of  the  entire  course  was  under  an  execu- 
tive committee  consisting  of  a  principal,  a  supervisor,  and  a  superin- 
dent  of  the  training  department  of  a  normal  school.  Each  subject 
was  in  charge  of  a  special  committee  consisting  of  teachers,  principals 
and  supervisors  with  the  teachers  largely  predominating.  While  the 
number  of  teachers  on  these  committees  was  made  as  large  as  possible 
in  order  to  secure  the  benefit  of  class  room  experience,  not  all  were 
able  to  participate  in  the  work  on  account  of  the  lack  of  time  and  faci- 
lities for  reaching  them.  Much  credit  is  due  all  who  have  so  will- 
ingly and  efliciently  assisted  in  bringing  this  course  of  study  to  its 
present  standard.  The  fact  that  it  is  an  outgrowth  of  the  best  class 
room  practice  in  the  city  is  due  largely,  however,  to  the  teachers  who 
helped  in  its  construction. 

The  general  i)lan  for  each  subject  in  the  Course,  the  i)rinciples 
for  the  selection  of  subject  matter,  and  the  organization  of  subject 
matter  were  agreed  upon  by  the  executive  coiniiiittee  and  the  chair- 
man of  each  special  committee  after  much  study  and  careful  delibera- 
tion. Each  special  committee  observed  these  principles  of  selection 
and  plan  of  organization  in  preitaring  the  subject  assigned.  Sugges- 
tions on  the  Course  in  English  were  received  from  a  grouj)  of  business 
men  in  order  to  secure  the  i)oiut  of  view  of  those  outside  the  schools. 
Similar  help  was  received  from  a  group  of  musicians  on  the  Course 
in  !Music. 

The  general  jjlan  adopted  for  each  course  is  as  follows: 

I.  Table  of  Contents. 

II.  Aiius  and  purposes  for  all  grades. 

A  statement  of  the  pur])oses  of  the  subject  as  a  whole. 


III.  Outline  oi  8uV>je(.'t  matter. 

IJrief  survey  of  subject  matter  throughout  the  Elemen- 
tary and  Junior  High  Schools. 

IV.  General  Directions. 

V.  Detailed  outline  of  subject  matter. 

VI.  General  Bibliography. 

As  a  basis  for  the  selection  of  subject  matter  for  this  Course  of 
Study  the  following  social  values  were  used: 

I.  That  subject  matter  was  selected  which  is  most  frequently 
used  by  the  greatest  number  of  people  in  life  situations.  The  term 
"use"  is  not  restricted  to  the  mere  economic  sense  but  includes  all 
those  matters  which  society  has  learned  to  value  and  desires  to  pass 
on  to  the  next  generation. 

II.  That  subject  matter  was  selected  which  is  not  only  most 
fretjuently  used  but  is  most  significant  when  used,  e.g.  we  teach  how 
to  save  life  from  drowning  not  because  of  the  number  of  times  it  would 
be  used  but  because  of  its  great  significance  when  used.  These 
method.'*  of  choosing  subject  matter  while  they  have  l)een  a  guiding 
principle  have  been  necessarily  limited  by  such  considerations  as,  ex- 
pense of  teaching,  time  of  i)uijils,  ability  of  teachers  and  pupils  and 
organization  and  availability  of  material. 

In  the  organization  of  subject  matter  an  attempt  has  been  made 
to  arrange  it  around  projects  suited  to  the  abilities  and  interests  of 
the  pupils  for  whom  it  is  intended  and  adapted  to  the  successful  use 
of  well  recognized  methods  of  teaching  and  to  the  needs  of  the  state 
and  community.  These  projects,  according  to  the  nature  of  the  sub- 
ject matter,  lend  themselves  to  one  of  the  following  types: 

Type  1. 

"lu  which  the  purpose  is  to  embody  some  idea  or  plan  in  exter- 
nal form,  as  building  a  boat,  writing  a  letter,  presenting  a  play : 

Type  2. 

"In  which  the  i^urpose  is  to  enjoy  some  aesthetic  experience,  as 
listening  to  a  .story,  hearing  a  symjjhouy,  appreciating  a  picture: 

TvPK  8. 

In  which  the  purpose  is  to  straighten  out  some  intellectual  dif- 
ficulty, to  Holve  some  problem,  as  to  find  out  whether  or  not  dew  falls, 
to  ascertain  howNew  York  outgrew  Philadelphia. 

TvPK  4. 

In  which  the  jiurjid.'^e  is  to  obtain  some  item  or  degree  of  skill 
or  knowledge,  as  learning  to  write  graile  14  on  the  Thorndike  Scale, 
learning  the  irregular  verbs  in  French.     .     .       .       Some  teachers  in- 


Ill 

deed  may  not  closely  discritniuate  between  drill  as  a  project  and  drill 
as  a  set  task,  althouffh  the  results  will  be  markedly  different." 

"It  is  at  once  evident  that  these  jjroupinffs  more  or  less  overlap 
and  that  one  type  may  be  used  as  means  to  another  as  end.  It  may 
be  of  interest  to  note  that  with  these  definitions  the  project  method 
logically  includes  the  problem  method  as  a  special  case.  The  value 
of  such  a  classiiicatiou  as  that  here  given  seems  to  me  to  lie  in  the 
ligfht  it  should  throw  on  the  kind  of  projects  teachers  may  expect  and 
on  the  procedure  that  normally  prevails  in  the  several  types." — 
Kilpatriek.     Teachers  College  Record,  September,  1918. 

This  Course  of  Study  is  in  no  sense  a  finished  product.  It  is  a 
record  of  past  achievement  and  a  standard  of  present  attainment.  It 
is  intended  also  to  be  a  guide  post  for  further  progress.  As  the 
quality  of  the  class  room  instruction  improves  by  means  of  this  course, 
the  course  should  likewise  be  improved  in  the  nature  of  the  subject 
matter  and  in  the  effectiveness  of  the  teaching  method.  For  this 
purpose  the  suggestions  and  criticisms  of  teachers,  principals,  and 
supervisors  will  be  requested  from  time  to  time. 


ARITHMETIC. 


AIMS. 


To  euable  one  to  do  llie  onliiiary  coiiiputiiiy  re(iuii-eil  in  cormnoii 

busiuess. 
To  give  arithmetical  kiiowledye  that  tit.s  into  the  real  situations 

of  home,  shop,  farm,  or  business. 
To  acquaint  pupil  with  the  simplest  and  shortest  methods  used  iu 

busiuess. 

ATTITUDES  TO  BE  DEVELOPED. 

"a  tendency  not  to  be  satisfied  with  guessing  or  approximation, 
but  to  insist  on  finding  out  through  the  iise  of  figures  all  essential 
matters  involving  numerical  values. 

"Standards  of  busiuess  accuracy  that  will  result  in  the  keeping  of 
an  accurate  account  of  all  personal  or  household  receipts  and  expendi- 
tures. This  will  make  possible  a  proper  adjustment  of  expenditure 
to  income  and  also  a  right  balance  among  the  different  objects  for 
which  money  is  spent. 

"Unwillingness  to  rely  on  general  estimates  or  rough  approxima- 
tions with  reference  to  projects  planned,  as. improving  a  home  or  a 
farm,  taking  a  trip,  investing  in  an  automobile,  etc. 

"Insistence  on  detailed  and  accurately  kept  records  of  profits  or 
losses  from  the  different  enterprises  of  farm,  shop,  or  busiuess. 

"The  development  of  such  a  sense  of  values  aud  the  inevitaVde 
logic  of  figures  as  will  render  one  proof  against  the  get-rich-quick 
schemes  ]»lanned  by  unscrupulous  i)romoters  to  catch  those  who 
tlirough  ignorance  of  Vjusiness  believe  wealth  to  be  attained  V)y  some 
kind  of  magic. 

'a  sense  of  pleasure  and  satisfaction  in  the  use  of  figures  and  in 
the  certainty  which  comes  from  their  wise  application  to  one's  affairs." 
Hetts:  Classroom  ^Methods  and  Management. 


BRIEF  SURVEY  THROUGHOUT  THE  GRADES. 

(rKAI)K  I. 

Xum])er  work  in   this  grade  is   taught  inciueutally  wliere  need 
arises  in  games  or  activities. 

GRADE  II. 

NOTATION  AND  NUMERATION.     Counting,  rea<liug  aud  writing 
numbers  to  100. 

ADDITION.  The  ad<lition  combinations  iu  which  the  sum   is  less 
than  10:  svmV)ols  +  aud  ^=. 


SUBTRACTION.    The  reverse   of  the  adilition  combinations:  the 

sytiihdl  — . 

MEASURES.  The  (juarter.  half-(luHar,  yallou,  ijouiid,  dozen,  half- 
•  lozen.  huuse  miiiiln'r.  the  date. 

(4KADE  II— A. 

NOTATION  AND  NUMERATION.  Counting,  reading  and  writing 
mimV>ers  to  200.   IJoman  numerals  to  XII.   Counting  by  3's,  4's,  and  ti's. 

ADDITION.  The  average  and  hard  combinations.  Column  ad- 
dition. 

SUBTRACTION.      The  reverse  of  the  addition  combinations. 

MULTIPLICATION.  The  tables  of  2,  5,  and  10.  The  symbol  X  as 
"times."' 

DIXISION.  The  reverse  of  the  multiplication  tables.  The 
symbol-^. 

MEASURES.  The  names  and  number  of  the  days  of  the  week. 
The  names  and  number  of  the  months  in  the  year.  The  hours  on  the 
clock  face.     The  signs  §  and  c. 

FRACTIONS.   2,  3  and  5  of  numbers  exactly  divisible. 

GRADE  III— B. 

NOTATION  AND  NUMERATION.  Reading  and  writing  numbers 
to  5000.   Roman  numerals  to  XX.     Counting  by  2  and  4  to  100. 

ADDITION.  The  very  hard  combinations.  Two  and  three  place 
column.s  iuvidving  carrying;  not  over  tive  addends.  Mixture  of  one, 
two,  and  three  jdace  columns.  Drill  on  endings. 

SUBTR.^CTION.  Easy  subtraction  and  one  step  borrowing. 
Work  on  entlings. 

MULTIPLICATION.  The  tables  of  3  and  4.  Multiplication  by 
one-place  multipliers. 

DUISION.  Division  as  the  reverse  of  multiplication.  No  carry- 
ing.   The  long  division  brace. 

MEASURES.  The  ounce  and  yard.  Time;  the  hour,  half-hour,  and 
quarter-hour. 

FRACTIONS.  3  and  \  of  numbers  which  are  exactly  divisible. 
Addition  and  subtraction  of  very  simple  fractions,  involving  no 
reductions. 

(iRADE  III— A. 

NOTATION  AND  NUMERATION.  Counting  by  2's  to  84;  8's  to  96; 
9'8  to  lOS.  Reading  and  writing  numbers  to  100,000.  The  use  of  the 
comma  as  in  14,  5U7. 

ADDITION.  Adding  four  and  five  place  columns  of  six  addends. 
Rapid  drill.  Speed  and  time  tests. 

SUBTRACTION.    Two-step  borrowing. 

MULTIPLICATION.  The  tables  of  6  and  7.  Multiplication  by  1 
and  2  place  niultii)lierH,  the  jjroiluct  to  contain  less  than  six  digits. 

DIV'LSIO.N.  Short  <livision  with  borrowing.  Numbers  not 
exactly    divisil)le. 


DENOMINATE  NUMBERS.  Oue-half  and  one-quarter  iuch.  The 
thermometer. 

FRACTIONS.  I  and  ]  of  numbers  to  100,  iiicludiiiti'  numbers  not 
exactly  divisible.  Adding  and  subtracting  simple  fractions  and  mixed 
numbers. 

GARDE  IV— B. 

NOTATION  AND  NUMERATION.  Reading  and  writing  numbers 
to  1,000,000.  Counting  to  100  by  ll's  and  12's. 

ADDITION.     Drill.  Addition  of  columns   containing   7   addends. 

SUBTRACTION.  Three-step  borrowing.  Rapi<l  drill.  Speed  and 
time  tests. 

MULTIPLICATION.  Tables  of  S  and  9.  Multiplication  by  two  and 
three  digits.   Zero  in  the  multiplier. 

DIVISION.  As  the  reverse  of  multiplication.  Rapid  drill.  ISpeed 
and  time  tests. 

MEASURES.    The  ton,  peck,  and  bushel. 

FRACTIONS.  I  and  g  of  any  two  place  numbers.  Addition  and 
subtraction  of  simple  fractions  and  mixed  numbers  with  no  reductions. 

DECIMALS.     U.  S.  money.  The  correct  use  of  "and." 

GRADE  IV— A. 

NOTATION  AND  NUMERATION.  Counting  to  100  by  15,  20  and 
25.     Roman  numerals  L,  C  and  D. 

ADDITION  AND   SUBTRACTION.  Time  tests. 

MULTIPLICATION.  Tables  of  11  and  12.  No  multipliers  over 
four  places.     Drill  for  speed. 

DIVISION.  Long  division,  using  a  two  place  divisor.  Zero  iu 
the  divisor.  Drill  for  speed. 

MEASURES.  Recognition  of  the  square  inch,  square  foot,  and 
square  yard. 

FRACTIONS.  i\  and  i\  of  any  two  place  numbers.  Addition  and 
subtraction  of  fractions  whose  denominators  may  be  seen  by  in- 
spection. 

DECIMALS.  Addition  and  subtraction  of  U.  S.  money.  Multipli- 
cation and  division  of  U.  S.  money  by  integers  only. 

GRADE  V— H. 

FUNDAMENTAL  PROCESSES.      Drills  for    speed    and    accuracy. 
Division  by  three  i)lace  <li visors. 
FRACTIONS.   Reduction  to— 

a.  Higher  terms 

b.  Lower  terms 

c.  An  integer  or  mixed  number 

d.  Like  denominators. 


4 

ADDITIUN  AND  SUBTRACTION  OF— 

a.  Similar  fractious 

1).  Dissimilar  fractions 

c.   Mixeil  numbers 

<1.  Applicatiou  to  problems  of  the  chiliVs  exi)erieuce. 
MEASURES.    .Miuutes  iu  the  hour  aud  seconds  in  the  minute. 
DECIMALS.     Expression  as  decimals  of  fractions  having  denom- 
inators of  10,  100.  or  1.000.    The  decimal  point  in  U.  S.  money. 

GRADE  V— A. 

FUNDAMENTAL  PROCESSES.    Drills  for  speed  aud  accuracy. 
FRACTIONS.    Multiplication  aud  Division  of: 

Fraction  by  a  whole  uumber. 

Whole  numlter  by  a  fraction. 

Fraction  by  a  fraction. 

Mixed  uumber  by  a  whole  uumber. 

3Iixed  uumber  by  a  mixed  number. 

Application  to  problems  of  the  child's  experience. 
MEASURES.    Tables  of  linear,  dry  aud   liquid  measure,   avoirdu- 
pois, time,  aud  V.  S.  money. 

DECIALALS.     Drill  on  U.  8.  money. 

PERCENTAGE.  Aliquot  parts  of  a  dollar,  S  .25,  §  .50,  8  .75,  S  .10, 
8  .20.  8  .1*2  2,  aud  their  fractional  equivalents. 

GRADE  VI— B. 

DENOMINATE  NUMBERS.  Addition,  subtraction,  multiplica- 
tion, division,  aud  oue  step  reductions. 

DECIMALS.  Addition  aud  subtraction,  multiplication,  aud 
division,  not  over  four  places.     Notation  aud  numeration  to  .000. 

PERCENTAGE.     Very  simple  table. 


(iRADE  VI— A. 

DENUMINATE  NUMBERS. 

Two  step  reductions. 

Tables  of  Square  and  Cubic  Measure. 

Area  aud  perimeter  of  rectangle. 

Area  of  triangle. 

Terms:  Rase,  altitude,  perimeter,  aud  area. 

lioard  Measure. 

C'ajiacity  of  bins,  coal  hauling,  and  wood  selling. 

IJegin  simjde  three  step  problems. 

DECI.M.ALS.     Notation  and  numeration  to  six  places. 

PERCE.N'T.AGE.    Change  decimals  to   percents.    Contiuuation   of 
percentage  tables  from  VI— R. 


5 
GRADE  VII— B. 
PERCENTAGE. 

Commercial  Discount. 
Profit  and  loss. 
Commission. 
Budgets. 
Equations. 

GRADE  VII— A. 
PERCENTAGE. 

Taxes. 

Duties  or  Customs. 

Insurance:  Fire,  Marine,  Life. 

Interest:  Simple,  Compound. 
GRAPHING. 


GRADE  VIII— B. 

INVOLUTION.  Squares  of  numbers  to  twenty-five,  cubes  of 
numbers  to  twelve. 

EVOLUTION.  Square  root.  Meaning  of  power,  exponent,  square, 
square  root,  and  radical  sign. 

MENSURATION.  Areas  of  rectangle,  rhomboid,  trapezoid,  and 
triangle.  Circle,  circumference,  radius,  diameter,  arc,  and  sector 
defined.  Area. 


GRADE  VIII— A. 

MENSURATION.  Lateral  surfaces  as  applied  to  prisms,  cylind- 
ers, pyramids  and  cones.  Volumes  of  prisms,  cylinders,  pyramids 
and  cones.     Excavations,  brick,  stone  and  concrete  work. 

BANK  DISCOUNT. 

BUSINESS  ARITHMETIC.  Common  business  forms, bills,  receipts, 
statements,  bills  of  lading;  itersoual  accounts,  debit,  credit  and  bal- 
ance, inventories;  sending  and  investing  money. 


GRADE  IX— B. 

COMMERCIAL  ARITHMETIC.      Addition,    subtraction,   multipli- 
cation and  division  of  whole  numbers,  fractious  and  decimals. 
PERCENTAGE. 


GRADE  IX— A. 

COMMERCIAL  ARITHMETIC.    Banking,  Iu.-urance,  Commission, 
Customs,  Duties,  Taxes,  Stocks  and  Bonds,  Business  Forms. 


6 
GENERAL  DIRECTIONS  AND  DISCUSSION  OF  PRINCIPLES. 

Identical  Forms  for  all  Grades  and  Buildings. 

Ecouomy  requires  that  in  the  rnechanies  of  numbers  there  should 
he  uniformity  of  forms  and  processes  throughout  the  school  system  so 
that  individual  children  or  classes  upon  being  transferred  or  promo- 
ted may  not  lose  time  or  energy  in  changing  methods  of  manipulation 
to  ctmform  to  different  practices.  The  following  processes  suggest- 
ed in  |)art  by  the  C'onnersville  Mathematics  Course;  Brown  and  Coff- 
man:  How  to  Teach  Arithmetic,  and  recent  arithmetics,  are  recom- 
mende<l  for  use  by  all  pui)ils  ami  teachers  in  the  grades  in  which  they 
occur.  If  a  child  already  handles  other  correct  methods  easily,  it  is  a 
doubtful  jiolicy  to  require  him  to  change. 

ISE  VERY  SIMPLE  FORMS  forthe  statement  of  written  problems. 

55  books 
25  books^ 
~80  books 


IX  .ADDITION  always  begin  at  the  bottom  of  the  coUmm  in  add- 
ing, adding  in  what  is  carried.  The  child  says,  "Three,  seven,  twelve. 
Write  two  and  carry  one.  One,  five,  seven,  eight.  Write  eight.  Six^ 
ten,  thirteen.  Write  thirteen.  The  sura  is  1382."  Check  by  adding 
down.  This  is  advised  because  many  examples  in  arithmetics  are  ar- 
range<l  to  jtrovide  for  the  recurrence  of  certain  combinations  if  added 
from  the  bottom  uj). 

315 
424 
643 


1382 


I.\  SUBTRACTION  the  child  says,  "Seven  hundred  eighty-two  less 
hree  hun<lred  sixty-seven.  Take  one  from  eight  to  put  with  two  and 
make  it  twelve.  Seven  ami  five  are  twelve.  Write  five.  Six  and  one 
are  seven.  Write  one.  Three  and  four  are  seven.  Write  four. 
The  remainder  is  415." 

7S2 
367 
415 

IN  MILTIPLICATION.  the  child  says  "Five  6's  are  thirty.  Write 
zero.  Five  two's  are  ten  and  three  are  thirteen.  Write  three.  Five 
5'8  are  twenty-five  and  one  is  twenty-six.     The  product  is  2630. 

526 
5 

2<i3() 


IN  DIVISION,  the  child  says,"Six's  iu  fourteen,  two.  Two  G'sare 
12.  Six's  iu  28,  three.  Three  tl's  are  IS,  etc."  Use  the  same  form 
for  both  long  or  short  division. 

239  (>218 

6)1484  2)1248(5 

12       .  12 

23  ~~4 

18  4 

54  ~~8 

54  2 

16 
16 

IN  ADDITION  of  fractions  proceed  as  follows: 

1 4  o  1 o    4 

i=A  5i=5A 

l=r\  2|-2t\ 

10. 1     7  03  5 -  1  1 

12— lo  6ii— <ii 

IN  SUBTRACTION  of  fractions  proceed  as  follows: 

3 —  n 
T — T"2 
1 —  4 
^ — T2 


IN  MULTIPLICATION  of  fractions  proceed  as  follows: 
2       7       7 

38      12 
4 

IN  MULTIPLICATION  of  mixed  numbers,  proceed  as  follows: 

8 

15     13     89 
7i  X  2 1=—  X  —  =— = 1 9i 
2       -5       2 

IN  DIVISION  of  fractions  procee<l  as  follows: 

2 

5       4      10 
(a)  t-|=-X-=-=U 

639 
3 

11 

77      4      44 

(b)          25|-1|=-X— =-=14f 
3       7       3 


8 
11.       Ill   MLLTIPLICATIOX  of  (leciinal  fractious  proceed  as  follows: 


2.2  5 
2.5 

1125 
450 

5.6  25 

12.      In  DIVISION  of  decimal  fractions  proceed  as  follows: 

85U.2 
.25). ST. 55 

18.     In  simple  interest  proceed  as  follows: 

(a)      Interest  on  810U  at  6°=  for  4  years. 

m^       6        4 

X  —  X— =824  interest. 

1        100      1 

(1»)      Interest  on  8100  at  5    °  for  2  years,  6  mouths. 

8^00       5       30 

X  —  X— =  812.50  iuterest. 

1       ;00     12 

(c)      Interest  on  8100  at  6    °  for  1  year,  2  mouths,  15  days. 

87 
8,100      ()      ^35 

X  —  X  —  =87.25  interest. 

1       ;00     ^60 

(30 

12 

Checks. 

Although  it  is  true  that  accuracy  in  arithmetical  processes  is 
larfrely  the  result  of  loug,  continued  practice,  yet  eveu  accountants  of 
great  e.xiterience  admit  a  liability  to  error  and  seek  to  verify  their 
computations  hy  some  check.  Children,  too,  feel  a  strong-  desire  to 
know  whether  or  not  their  solutsons  are  correct  and  if  no  other  means 
than  the  <lecision  of  the  teacher  he  provided,  show  such  a  desire  by 
<|uesti()ns,  a«  "Is  my  answer  right?"  "What  part  is  wrong?" 

It  is,  then,  to  give  children  the  satisfaction  and  self  confidence 
which  comes  from  knowing  how  to  determine  whether  or  not  that 
which  one  does  is  correct,  and  to  satisfy  the  demand  of  the  business 
world  for  accuracy,  that  a  practical  metliod  of  verification  should  be 
taught.  After  such  a  check  is  once  taught,  the  child  should  use  it 
until  it  becomes  entiielv  habitual  f)r  automatic. 


9 

Some  of  the  checks  most  commouly  used  au<l  believed  to  he  of 
greatest  heuetit  in  the  elimination  of  errors  are  listed  helow: 
Estimatin<)-  results  to  avoid  absurd  errors. 
Estimating'  the  ai)j)roximate  field  in  which  the  answer  should 
lie  before  the  problem  is  attacked  is  very  helpful  in  doinj;  away 
with  ludicrous  errors   and   establishing  the  sense   of  numerical 
values. 

Ex.  A  man  earns  §2.50  a  day.  What  are  his  earning-s  in  a 
week?  The  child  at  once  sees  that  the  amount  of  his 
earnings  must  be  in  the  neighborhood  of  §12.  This  es- 
timation precludes  error  in  placing  decimal  i)oint. 

What  is  the  length  and  width  of  our  schoolroom? 
Children  should  judge  the  length  and  width  first  with 
the  eye,  and  then  compare  their  estimation  with  actual 
results. 

How  many  rods  long  is  the  walk  in  front  of  the 
school?  Children  estimate  the  distance  by  walking,  veri- 
fying later  by  actual  measurement. 

Checks  for  the  four  fundamental  operations. 

ADDITION:  Checked  by  adding  digits  in  the  reverse 
order.  Be  careful,  however,  in  adding  down,  that  new  and 
strange  combinations  are  not  encountered.  It  is  V)etter  to 
delay  the  iuti'oductiou  of  this  check  until  all  the  combina- 
tions have  been  learned. 

65  65 

24  24 

Add  uj)  iq  Add  down         ig 

13  13 

121  121 

SUBTRACTION:    Check  (l)  by  adding  difference  to  the 

■  subtrahend   to  form  the  minuend  or  (2)  subtracting   result 
from  minuend  to  obtain  subtrahend. 

Ex,                      Check  1  Check  2 

924                                689  924 

Subtract_235                       Add_235  Subtract  6S9 

689                                 924  235 

MULTIPLICATION:  Checked  (l)  by  dividing  product 
by  multiplicand  to  form  multii)lier  or  (2)l)y  <lividing  i)roduct 
by  multiplier  to  form  multiplicand. 

Check  1  Check  2 

Ex.  23  656 

656 

Multiply         23 
1968 
1312 
15U88 


65(5)  I  •'><»•'*>' 

23)15(IS,S 

1312 

138 

l!Mi8 

128 

19(iS 

115 

138 

138 

1(1 

DIVISION:  (iK-cki'il  li\-  inultiplyiua-  quotieut  and  divisor 
to  furiii  <livi<leii<l. 

4:{  Check 

1  :))()4.')  48 

)i()                       ^Iultii»ly  !•") 

45  •il-'^ 

45  48 


645 


Other  Checks. 

Kevifwiuy:  .stei).><  in  prolileni!5  iuvulviny:  small  uuml»ei's. 
For  reasuiiiiiff  problems: 

IJewordiug  to  clarify  purpose  of  problem. 
Ex.  (l)  How  much  more  does  coifee  at  25  lbs.  for 
>!9.50  cost  thau  coffee  at  33c  a  lb. 
(2)   IJewordiutj: 

One    grade  of    coffee  sells  at  §9.50  for  25 
lbs.     Another  grade  sells  for  83c.     Which 
costs  the  more? 
Statiug  the   same   type  of  problem  iu   simpler  nu- 
merical terms. 
Ex.  (l)  ^Ir.  Browu   wishes   to   buy   a    house.     He 
must    give   his    agent     10    per   cent  com- 
mission to  sell  a  home  worth  824,000.   How 
much  money  will  Mr.'  Browu  get  from  the 
sale? 
(2)    Reducing  to  simpler  numerical  terms: 

You  sell  strawberries  for  Mrs.  Brown.  She 
tells  you,  you  can  have  10  per  cent  of  the 
money  received  from  selling  the  berries. 
If  you  sell  S8.00  worth  one  morning,  how 
much  money  will  you  give  Mrs.  Brown  at 
the  close  of  the  morning? 
Solving  by  another  method: 

A  certain  store  takes  in  8100,000  <luring  a  year.  Its  ex- 
penses were  812,000  for  rent,  82.S,(JUU  for  salaries.  850,000  for 
stock  and  82,000  for  heat  and  light.  What  were  the  profits 
for  the  vear? 


Method    (1) 

(2) 

812,000 

8100.000 

50,000 

Subtract       12.000 

2.000 
2.s.0ll(i 

8  8S.00O 
Subtract       50,000 

892,000 

8   8S,000 

81(10.(101) 

Suljtract       2.S.0UO 

1)1!. (MMI 

8  10,000 

8     .S,00()  Piotit 

Subtract         2,000 

8     S,000   Profit 


11 

Inaccuries  to  Be  Avoided: 

See  JJiowTi  and  ('oftiiian:   IIow    to  Teaeli  ,\ritliMietic-. 
Inaccurate:  3+4=7+5=12 X2=24 
Accurate:       3+4=7;     7+5=12;     12 --2=24 

Inaccurate:  4|  =  i^.2  Accurate:  4|=4/2 

0-2 12  O  .,  —  6  1  2 


_L2_ 

ll  1 


9i^=10}i  lUl' 

Inaccurate:  2  >:$50=100  Accurate:  2X$50=$100 
2x50=8100  2-50=100 

8100^4=25  §100  :  4=!!i25 

100-^4=825  100  :  4=25 

8100-^84=825  8100  :  84=25 

Inaccurate:  4  ft.X5  ft. =20  sq.  ft. 
Accurate:      4/5  A 1  sq.  ft. =20  sq.  ft. 
4X5  sq.  ft.=20  sq.  ft. 

Inaccurate;  27  cu.  ft. ^9  sq.  ft. =3  ft. 
Accurate:      27  cu.  ft. -^9  cu.  ft. =3 

Problem:       Two-fifths    of  a    number  etiuals    12.      Find    the 

number. 
Inaccurate:    5  =the  number    Accurate:  5  of  the  nuiiiber=the 

number 
g=12  I  of  the  number  =12 

5=i  of  12=6  I    of   the   number=.', 

of  12=6 
|=.5X6=30  i    of    the     number= 

•    5X6=30 
Inaccurate:  100     per  cent=  Accurate:  100   per    cent    of  the 

100  number=the  number 

Inaccurate:  15°  =1  hour  of     Accurate:  15°  correspond    to  1 
time.  hour  of  time. 

Principles  of  method  in  teaching  Arithmetic  as  derived  form  scientific  in- 
vestigation from  the  18th  Year  Book  of  the  National  Society  for  the 
Study  of  Education. 

In  teachinfj  the  numi)er  concept,  which  is  fundamental  in  arith- 
metic, purposeful  experience  with  concrete  objects  should  be  i)rovided. 
Counting  and  measuriuy  are  two  of  the  most  fruitful  forms  of  experi- 
ence. 

In  jreneral,  the  meaning  should  be  taught  before  the  word  or  other 

symbol  is  given  to  the  child.     This  aj)plies  to  the  number  symbols   1. 

2,  3,  4,  etc.,  and   to  the  technical   words  of  arithmetic  such  as,  "athl." 

'subtract,"  "foot,'" 'V«ii"'li'  l'**"i"li'    J^^i"-'  "how  inucli."' and  the  like. 

Learning  the  tables  is  a  matter  of  memorizing  and  the  rules  for 
memorizing  ajiply. 

The  child  should   understand   the   meaning  of  the  comliina- 

tious  which  he  is  memorizing. 


12 

Attentive  rei>etitious  are  ueoessary  to  fix  the  associations  iu 

learniiifr  the  tables.     These  repititious   may  be  given    either  in 

ischitetl  ilrill  or  in  the  use  of  these  number  facts  in  the  doing  of 

exainjiles. 

For    jiermanent    memorizing:  [whit-li  i.s  desired  in  tliis  case} 

tlie  repetitions  of  drill  must    be  carrie<l   beyond   the  i)()int  where 

iminetiiate  recall  is  just  barely  jjossible. 

The  learning  should    be  done  under    some  pressure  or  con- 
centration. 

Memorize  the  number  facts  in  t^roups,  not  one  fact  at  a  time. 

In  learning  the  tables,  the  different  combinations  are  not  equally 
ditHcult  and  the  number  of  repetitions  of  the  several  combinations 
should  correspond  to  the  degree  of  difficulty,  the  most  difficult  receiv- 
ing the  largest  number  of  repetitions. 

Tlie  actual  degree  of  difficulty  of  a  combination  to  any  one  child 
is  an  "individual  iieculiarity".  This  condition  makes  it  necessary  to 
suitjilement  class  drills  by  i)rovisions  for  individual  practice. 

In  adilitioii  and  niultiplieatiou  l)oth  forms  of  each  combination 
should  be  taught. 

In  column  addition,  grouping  digits  to  make  10  or  some  other 
convenient  number  is  not  helpful.  A  variety  of  procedure  is  to  be 
expected,  and  the  best  results  are  obtained  when  pupils  are  urged  to 
work  rajiidly  but  are  allowed  to  choose  their  own  methods. 

The  Austrian,  or  additive,  niethod  of  subtraction  is  not  superior 
to  the  "take-away"  method. 

In  "borrowing"  it  is  better  to  ijicrease  the  subtrahend  by  one 
than  to  decrease  the  minuend. 

The  Austrian,  or  multiplicative  method  of  division  is  superior  to 
the  direct  association  method  in  the  initial  stages  of  learning. 

The  Austrian  method  of  placing  the  decimal  point  in  the  division 
of  decimals  is  more  efficient  than  the  traditional  method. 

Most  errors  belong  to  recurring  types.  The  most  frequent  of 
these  types  should  receive  special  em])hasi8  so  that  they  can  be  elim- 
inated. 

Pupils  should  be  taught  to  use  an  abbreviated  phraseology. 

After  the  initial  stage  of  practice,  drill  upon  the  fundamental 
combinations  should  be  given  by  means  of  examples. 

The  ijeriod  of  practice  should  be  from  10  to  15  minutes. 

Children's  knowledge  of  their  previous  performances,  combined 
with  the  desire  to  surpass  those  records,  is  the  greatest  factor  con- 
tributing to  imj>rovement. 

A  preliminary  practice  of  five  minutes  at  the  beginning  of  the 
recitation  serves  as  a  "mental  tonic." 

In  the  ojierations  of  arithmetic,  emphasis  should  be  i)laced  upon 
rapi<l  work  rather  than  ui)on  accuracy. 

I'upils  Itelonging  to  the  same  class  have  been  shown  to  differ 
widely  in  achievement.  This  condition  makes  it  necessary  to  provide 
f<ir  individual  instructions, 

'Fhe  Courtis  Standard  Practice  Tests  are  superior  to  Thompson's, 
miiiiinum  essentials. 


13 

Arithmetical  abilities  are  specific,  and  explicil  traiiiiiiir  iimst  be 
provided  tor  each  one. 

The  development  of  the  abilities  of  a  pupil  to  do  different  types 
of  examples  is  frequently  not  uniform.  To  meet  this  situation,  dia<r- 
uosis  and  corrective  instructions  are  required. 

Arithmetical  study  and  ])ractice  which  is  motivated  by  "itractical" 
!)roblems  produces  results  sui)erior  to  those  secured  by  usiii',''  tlie 
problems  in  the  textbook. 

[Systematic  measurement  of  the  results  of  teaching  l)y  standard- 
ized tests  produces  a  hijiher  degree  of  efficiency. 

Pupils  need  to  be  taught  the  meaning  of  technical  terms  that  are 
used  in  the  statement  of  problems,  as  a  prerequisite  for  reasoning  in 
solving  problems." 

Provision  for  Individual  Differences. 

The  usual  class  instruction  in  arithmetic  does  not  meet  the  needs 
of  individuals.  Classes  conducted  in  a  manner  similar  to  those  de- 
scribed by  Monroe  are  all  too  common:  "Frequently  the  writer  has 
visited  classes  in  arithmetic  which  were  being  drilled  upon  the  fund- 
amental operations.  A  fairly  uniform  procedure  was  followed.  The 
same  example  was  dictated  to  all  of  the  pupils,  regardless  of  whether 
they  needed  drill  upon  this  particular  type  of  example  or  not.  Nat- 
urally some  pupils  finished  very  quickly,  and,  as  they  waited  for  their 
classmates  to  finish,  there  was  a  tendency  for  them  to  become  dis- 
orderly—a perfectly  natural  tendency.  When  a  majority  of  the  class 
had  finished  the  example  the  teacher  stopped  the  work  and  read  the 
correct  answer.  The  process  was  then  repeated.  The  result  was 
that  those  pupils  who  worked  slowly  completed  few,  if  any,  examples 
during  the  entire  period,  and,  therefore  received  little  satisfactory 
drill.  The  bright  pupils  spent  a  considerable  proportion  of  their 
time  waiting  on  the  other  members  of  the  class,  and  i)robal)ly  did  not 
need  the  particular  kind  of  drill  which  they  received." 

lie  gives  the  following  suggestions  as  to  how  this  condition  may 
be  remedied:  "instead  of  dictating  only  one  example  at  a  time,  the 
teacher  can  dictate  several,  and  stojj  the  work  as  soon  as  a  few  of  the 
faster  workers  have  finished.  The  slow-working  pupils  will  have 
some  examples  finisned. 

"The  teacher  must  recognize  that  the  rate  at  which  the  jMipil 
performs  the  o])erations  is  important,  as  well  as  the  accuracy.  This 
means  that  in  teaching,  the  teacher  must  obtain  a  measure  of  the 
pupil's  speed,  as  well  as  a  measure  of  his  accuracy.  If  examjtles  are 
dictated  in  groui>s,  and  the  work  stoi)ped  as  suggested  in  the  above 
paragraph,  the  number  of  examples  which  the  \ni\)i\  does  during  llu* 
class  period  is  a  measure  of  his  rate  of  working.  The  i>er  cent  cor- 
rect is  a  measure  of  his  accuracy. 

"The  instruction  can  be  made  still  more  eftecti\c  it  the  teacher 
will  prei)are  a  numl)er  of  sets  of  exami)les,  each  set  being  confined  to 
examples  of  the  same  type.    These  sets  of  exaniiiles  should  be  written 


u 

oil  t-anls.  Then,  instead  of  dictatiuff  examples,  the  teacher  can  dis- 
tril»ute  tlie  cards  and  have  the  pupils  copy  the  examples  from  the 
car.ls.  If  the  teacher  studies  the  needs  of  her  pupils,  it  will  be  pos- 
sihle  fi>r  her  to  distribute  the  cards  so  that  each  pupil  will  have  the 
type  of  example  ujiou  which  he  needs  practice.  The  pupil  is  probably 
injured  liy  beini^  i-e(|uire<l  to  practice  upon  the  wrong  type  of  example 
and.  hence,  it  is  very  imjiortant  that  each  i)ui)il  be  gfiven  the  type  of 
examiile  upon  which  he  needs  practice."  Educational  Tests  and 
^leasurements.   ]».  .t.i. 

Although  problem  work  cannot  be  made  as  definitely  individual, 
still,  a  careful  classitication  should  l)e  made  of  failures:  children  with 
a  lack  of  knowledge  of  the  processes;  those  who  need  drill;  those  who 
lack  ability  in  reasoning  or  computation.  The  proper  remedial  mea- 
sures may  then  be  taken  by  devoting  some  of  the  class  time  for  special 
\vi>rk  with  the  above  mentioned  groui)S. 

"In  aiming  to  secure  good  quality,  the  teacher  will  not  allow  the 
subject  to  take  i)recedence  in  his  mind  over  the  jtupils.  A  true  teacher 
is  never  merely  teaching  a  subject.  lie  is  always  assisting  a  human 
being  by  means  of  a  subject  to  grow  and  adapt  himself  to 
his  surroundings.  It  is  then,  after  all,  the  pupils'  interest,  success, 
growth,  improvement  that  the  teacher  has  in  mind  when  he  sets  up 
high  stan<lards  and  puts  into  operation  stimulating  incentives. 

If,  however,  these  produce  failure  and  discouragement  in  any 
in<Iividual.  the  standards  and  incentives  are  surely  not  accomplishing 
the  most  useful  results.  Indivi<luals  and  classes  differ  in  their  mathe- 
matical aptitude.  They  come  to  a  teacher  with  different  kinds  of 
jireparation.  If  instruction  is  to  be  given  from  the  standpoint  of  the 
l»upils,  it  must  first  be  <liscovered  what  each  individual  knows  and 
also  how  he  knows  it.  Therefore,  standards  should  be  adjusted  to 
each  class  and  to  each  individual.  For  the  more  able  pupils  special 
advanced  problems  should  be  assigned  or  offered  as  optional  work  for 
which  special  credit  is  given.  For  the  less  able,  special  easier  work, 
should  ]je  arranged  to  secure  the  needed  review  and  drill."  Ken<lall 
an<l  .Mirick:  "How  to  Teach  the  Fundamental  Subjects." 

( )nly  l)y  eradicating  individual  weaknesses,  will  the  class  show 
the  greatest  amount  of  progress,  for  it  is  usually  only  a  few  members 
of  the  gr()U]>  who  ])ull  down  the  whole  class  standard.  (Graphing  and 
explaining  the  individual  and  class  results  in  standard  tests  is  a  won- 
derfid  stimulant  of  individual  effort.  The  degree  to  which  personal 
needs  are  met  is  tlie  measure  of  educational  efficiency. 

Standard  Practice  Material. 

Speed  and  accuracy  in  the  funihunental  ojierations  may  be  ac- 
«iuire<l  oidy  by  means  of  fre<iuent  drill  projects.  The  fundamentals 
should  have  the  right  of  way  in  any  grade.  In  the  upper  grades  the 
abilities  of  the  pui>ils  to  <lo  different  types  of  exami)les  are  frecpiently 
not  uniform.  To  meet  this  sitiiation  every  teacher  of  arithmetic  in 
grades  five  to  eight  should  check  the  abilities  of  incoming  groups  in 
tlie  four  fuiidaiiiental  oi>erations  and  diagnose  the  deficiencies  of  each 
chilrl  by  the  use  of  the  standard  tests  of  Woody  or  Courtis.  Definite 
practice  material  shouM  be  use<l  to  correct  conditions  discovered. 


15 

Two  sets  of  praftice  material  which  serve  the  imrixise  of  testiug 
and  ])ractit'iut):  in  l»oth  speed  and  accuracy  are: 

Courtis  Standai-d  l*ractice  Tests,  C'anI  Cabinet  FMition,  Cabinet  1, 
World  IJook  Company.  2126  Prairie  Avenue,  Chicafro. 

Studebaker  Economy  Practice  Exercises,  Set  B-One,  Scott- Fors- 
niau  and  Company,  528  South  Wabash  Avenue,  Chicatjo. 

These  are  particularly  valuable  in  grades  five  and  six.  A  few 
minutes  should  be  yiven  to  this  work  every  day,  each  child  at  his  own 
rate.  An  individual  may  be  excused  for  other  activities  when  he 
reaches  the  grade  standard.  These  i)ractice  tests  in  themselves  con- 
stitute the  true  drill  project  material  with  all  the  characteristics  of 
'  puri)osefur'  activities  when  correctly  used. 

Children  should  be  lead  to  note  their  progress  during  a  given  in- 
terval of  time  between  two  tests  rather  than  any  one  unrelated  score. 
Team  work  and  graphing  of  scores  will  emphasize  this. 

At  the  cdose  af  each  semester,  standard  tests  in  the  fundamental 
operations  should  be  given. 

Standard  Tests  in  Arithmetic. 

THE  WOODY  ARITHMETIC  SCALES. 

Woody  has  organized  a  set  of  four  scales,  one  for  each  of  the 
fundamental  processes.  His ''fun<lamental  idea  was  to  derive  a  series 
of  scales  which  would  indicate  the  type  of  problems  (exam])les)  and 
the  ditticulty  of  the  ]»rol)lems  (examples)  that  a  class  can  solve  cor- 
rectly  The  ditticulty  of  each    exami)le    has    been    determined 

and  the  examples  of  each  scale  are  arranged    in   oi'der  of    increasing 

difficulty The  score  of  a  pupil  is  a  statement  of  the  particular 

examples  which  he  has  done  correctly.  The  score  of  a  class  is  the 
degree  of  difficulty  of  the  example  which  was  done  correctly  by  just 
50  per  cent  of  the  class.  Ability  as  measured  by  these  scales  means 
simply  that  certain  types  of  examples  can  be  done  correctly  and  that 
certain  other  types  cannot  be  done  correctly.  The  speed  at  which  the 
examjdes  can  be  done  is  not  includeil  in  the  meaning  of  ability." 
Monroe:    Educational  Tests  an<l  Measurements,  p.  29. 

Following  are  the  medians  as   given   by  Wuddy   foi'  these  scales: 


SEIMKS   A  —  ACCrUACV. 

GRADE  ADDITION  SUBTRACTION  MULTIPLICATION  DIVISION 

Second 6.S  5.1                   

Third 14.5  11.2  4.7  5.S 

Fourth IS.B  15.()  11.1  9.S 

Fifth 28.1  2U.4  1S.8  16.4 

Sixth 29.7  24.9  26.1  28.7 

Seventh 82.4  2S.5  8U.0  27.4 

Eighth 88.9  81.7  82.9  :!(».! 

COURTIS  STANDARD  RESEARCH  TEST. 

This  series  consists  of  four  tests,  printed  on  tour  consecutive 
liages.  They  are  suitable  for  a  general  survey  of  the  abilitii-s  of  pujtils 
to  perform  the  operations    with  integers.     'I'he  following  are  sami>les: 


16 

TEST  NO.   TWO SUBTRACTION. 

"This  test  cousists  of  tweuty-four  examples,  each  iuvolviuo-  the 
same  miniber  of  subtractions.  The  followiug  are  samples.  Time  al- 
lowed. 4  minutes. 

10779.1491  750SSS24  91500053  S7939983 

77197»i-i9  ■')74(M>394  1!)9(il5(>3  7-22U731H 

TKST  No.  THKEE  — MILTII'LICATION. 

This  test  consists  of  twenty-four  examples  of  this  type.  Time  al- 
lowed, 6  minutes. 

S246  3597  5739  2648  9537 

29  73  85  46  92 

In  marking  the  test  papers,  wliich  is  done  by  the  use  of  a  printed 
answer  card  wliich  is  run  along  across  the  page,  no  credit  is  given  for 
examjdes  i)artly  right  nor  for  examples  partly  completed.  A  pupil's 
score  is  the  numljer  of  examples  attempted  and  the  uumlier  right. 
This  simple  plan  of  marking  the  papers  insures  uniformity  and  ac- 
curacy. 

Each  of  the  examples  of  a  test  calls  for  the  same  number  of 
operations  under  approximately  the  same  conditions.  This  makes 
the  examples  of  each  test  approximately  eq\ial  in  difficulty.  Any  ex- 
ample of  the  addition  test,  say  the  seventh,  is  just  as  difficult  as  any 
other,  say  the  second.  Thus,  the  tests  consist  of  twenty-four  equal 
units,  just  as  a  yardstick  consists  of  thirty-six  equal  units  (inches). 
The  measure  of  a  pupil's  ability  is  represented  by  the  distance  he  ad- 
vances along  the  scale  in  the  given  time,  i.  e.,  by  the  number  of  ex- 
amples done  and  by  the  per  cent  of  these  examples  which  have  been 
done  correctly. 

Since  an  example  of  one  of  these  tests  is  defined  as  so  many 
operations  under  certain  conditions,  it  is  possible  to  construct  other 
tests  equal  in  difficulty.  Four  forms  have  been  constructed.  This 
makes  it  possible  to  use  a  different  form  when  the  tests  are  given  a 
secontl  time.'"   Monroe:  Educational  Tests  and  Measurements,  p.  23-25. 

These  are  the  medians  obtained  by  Courtis: 

SEKIES  B — SPEED. 
GRADE  ADDITION  SUBTRACTION  MULTIPLICATION  DIVISION 

Third 4  5 

Fourth 6  7  6  4 

Fifth 8  9  8  6 

Sixth 10  11  9  8 

Seventh 11  12  10  10 

Eighth 12  13  11  11 

SERIES  B  —  ACCUEACV. 
<5RADE  ADDITION  SUBTRACTION  MULTIPLICATION  DIVISION 

Third 41  49  

Fourth 64  80  67  57 

Fifth 70  83  75  77 

.Sixth 73  85  78  87 

.Seventh 75  86  80  90 

Eighth _.  76  87  81  91 


17 
'THE  STONE  REASONING  TEST. 

"Stoue  has  worked  out  a  reasoniufj;'  test  which  has  heen  used  in 
several  cities,  and  in  a  number  of  city  school  surveys,  so  that  we  have 
rather  definite  standards  as  to  Avhat  may  l)e  expected  from  its  use." 

ACTUAL  MEDIANS  OBTAINED. 

GRADE  Butte,      Bridgeport,    Salt  Lake       Nassau,  Lead, 

1914.    Conn.,  1913   City,  1915.  Co.,  NY,  1918.  S.  Dak.,  1916 

Fifth 2.2  t).l  3.7  ___  4.5 

Sixth 3.9  5.2  (j.4  4.5  6.1 

Seventh 5.8  6.8  8.6  __.  9.3 

Eighth 7.7  4.5  10.5  8.2  11.4 

Tentative  Standards  suggested  by  Stone  (1916)  for  his  Reason- 
ing Tests: 

That  80  per  cent  or  more  of  5th  grade  pupils  reach  or  exceed  a 
score  of  5.5  with  at  least  75  per  cent  accuracy. 

That  80  per  cent  or  more  of  6th  grade  pupils  reach  or  exceed 
a  score  of  6.5  with  at  least  80  per  cent  accuracy. 

That  80  per  cent  or  more  of  7th  grade  pupils  reach  or  exceed  7.5 
with  at  least  85  per  cent  accuracy. 

That  80  per  cent  or  more  of  the  8th  grade  pupils  reach  or  exceed 
a  score  of  8.75  wnth  at  least  90  per  cent  accuracy. 

The  time  allowance  for  the  test  is  fifteen  minutes.  Stone's  plan 
for  marking  the  test  papers  allows  credit  for  examples  partly  right 
and  for  examples  which  are  not  finished.  The  problem  values  have 
been  determined  upon  the  basis  of  difficulty.  It  should  be  noted  that 
this  plan  for  marking  the  test  papers  is  not  as  simple  as  that  em- 
ployed for  marking  the  test  papers  on  the  operations  of  arithmetic." 
3Ionroe:  Educational  Tests  and  Measurements,  p.  36,  37. 


IS 

(iKADK  I-J}  au.l  I-A 
DIRECTIONS. 

DIVISION  OF  TIME— All  the  arithmetic  work  of  the  t^-rade  will 
l>e  oral.  Only  such  writin<r  of  luunbers  as  is  required  ineideiitally  in 
the  seiirintr  of  y:aines  neeil  l)e  done.  All  the  work  of  the  {ji'ade  will 
lie  eonerete. 

The  I-B  grade  will  have  no  arithmetic  work  except  iuei<lentally. 
Do  not  give  a  separate  period  to  the  teaching  of  numbers  in  the  I-A 
grade  hut  teach  the  numbers  in  connection  with  the  other  work. 

The  following  ways  of  teaching  the  number  concept  are  recog- 
nized as  best: 

Relating  the  number  work  to  the  chihrs  experience  thru  the  use 
of  objects.  Dr.  Suzzalo  in  his  Teaching  of  Primary  Arithmetic,  pp. 
49-5U.  makes  a  plea  against  the  very  narrow  range  of  objects  used  in 
this  work,  or  to  (piote  exactly:  "It  is  too  frequently  the  case  that  teach- 
ers will  treat  the  fundamental  combinations  with  one  set  of  objects, 
e.  g.  lentils.  In  all  the  objective  experience  within  that  field  there 
are  two  persistent  associations,  'lentils,'  and  the  relation  of  addition. 
It  follows  then,  that  in  the  effective  use  of  this  method  the  range  of 
material  should  be  as  wide  as  the  teacher's  ingenuity  can  make  it. 

Through  rhythm.  Songs  which  have  marked  rhythm  may  be 
claitped.  tajtited.  or  counted. 

Through  games,  drawings,  and  construction  work. 

SUBJECT   MATTER   AND    PROJECTS. 

COUNTING:  l>y  I's  incidentally  where  need  may  arise  in  games 
or  activities. 

Projects:  To  count  pencils  or  crayons  for  the  class;  erasers,  seats, 
chairs,  papers  needed  for  each  row;  pictures  on  the  wall;  beau  bags; 
boys  in  the  room;  girls  in  the  room;  boys  and  girls  absent;  etc. 

To  count  ai)i)les,  beads,  objects  of  different  colors  and  shapes, 
inch  Sfpiares  or  heavy  cardl)oard  strips  cut  from  bristol  board,  toy 
money,  jjinl  and  quart  measures,  circles,  and  speer  blocks  of  various 
forms  and  sizes. 

To  jilay  Hop  Scotch,  Dominoes;  IJeast,  Iiird,  and  Fish,  etc. 

To  teach  the  sense  of  number  thru  the  medium  of  play. 

To  tell  the  pages  and  lessons  in  the  readers. 

MEASURES:  Cent,  nickel,  dime,  dollai-,  inch,  foot,  pint,  (piart., 
and  recognition  of  circle  and  square. 

Projects:   liec(»gnize  these  measures  in  games: 

Play  store. 

Measure  the  sand  table  and  the  sand  in  it. 

Construct  siinjile  objects  suggested  by  the  industrial  arts  outline. 

Optional  Work. 

Hea<ling  and  u  riling  numbers  to  50. 
(Vninting  biicku  ai-ds  by  I's  from  50. 


19 

Standards  of  Attainment. 

See  sul»jeft  matter. 

Bibliography. 

J>r(i\vii  and  ("offinau:  ITow  to  Teach  Aiitliiiietic.  C'bai).  X.  Pri- 
mary Arithmetic. 

Johusou:   Education  l)y  Flays  and  (iames. 

Klapper:  The  Teaching  of  Arithmetic,  ('liaji.  \'I.  Teaching- tlie 
Number  Concept. 

McMurray:  Special  Metho<l  in  Arithii.etic.  C'hai*.  III.  ^letliod 
in  Primary  Grades. 

Suzzalo:  The  Teacliinjg:  of  Primary  Arithmetic.  C'lia]).  \.  ami  \]. 
Objective  Work. 


(;ka1)K  ii-r. 

DIRECTIONS. 

DIVISION  OF  TIME:  Nearly  all  work  should  be  oral.  Three- 
tourtlis  of  the  work  in  this  trraile  shoiiM  Ite  coiu-rete. 

REVIEW.     Measures  aii<l  c•ouIltillL^ 

MAIN  TOPICS:  Counting  u].  to  100.  addition  and  subtraction 
eoinbinations,  niea.sures. 

NOTATION  AND  NUMERATION.  In  reading  numbers,  do  not  al- 
low the  ehiltlren  to  misuse  and.  '245  should  be  read, "two  hundred 
t\>rty-tive"  not  "two  liuiidred  and  torty-tive."  And  indicates  a  deci- 
mal point.  This  caution  apidies  also  to  the  reailiufj:  of  the  date.  See 
IV-I>  Directions. 

ADDITION  AND  SUBTRACTION.  In  all  types  of  work,  omit  tlrill 
ou  facts  and  processes  fully  known,  and  stress  those  coml)inations  or 
operations  of  which  the  class  is  not  sure.  Provide  special  help  for 
individuals  to  overcome  their  ]»articular  weaknesses.  This  involves 
frequent  testing  to  discover  both   class  and  indiviilual  shortcomings. 

Make  the  combinations  automatic,  thru  memory  work,  permitting 
no  counting  after  the  lirst  experience. 

Test  the  pupils  frequently  on  the  combinations  committed  to 
memory.  Do  not  permit  the  child  to  hesitate  iu  giving  these  combin- 
ations. If  he  does  so,  it  is  better  to  state  the  result  or  call  at  once  on 
another  ])upil. 

Use  {\)e  column  form  only  iu  teaching  addition  and  subtraction, 
avoiding  the  use  of  plus  and  minus  signs.  Teach  the  recognition  of 
these  signs  only. 

Teach  the  addition  and  subtraction  combinations  simultaneously 
using  the  Austrian  or  "change  making"  methoil  iu  subtraction  for  the 
sake  of  uniformity  over  the  city.      See  page    21. 

Smith  iu  the  Teachers'  College  Record  1909,  p.  46  says:  "The 
addition  consists  iu  finding  what  number  must  be  added  to  the  subtra- 
hend to  make  the  minuend.     Thus  in  thinking  of  17 — 8  we  think,     8 

ami  9  are  17,"  writing  down  9 Is  the  general  plan  the  best 

one?  On  the  side  of  advantages  we  have:  (l)  It  is  the  common  meth- 
od of  making  change (2)  It  avoids  the  necessity  of  making  a 

separate  subtraction  table.  There  is,  therefore,  an  economy  of  time 
and  an  increased  efficiency  in  the  very  important  subject  of  aildition. 
(y)  The  facts  of  addition  being  used  so  much  md^  often  than  those  of 
subtraction  there  is  naturally  an  increase  in  speed  and  certainty  when 
we  emjiloy  the  addition  instead  of  the  subtraction  tal)le." 

Avoid  study  jicriods  which  fix  l)ad  habits. 

Play  games  for  the  game's  sake,  V)ut  avoiil  games  which  are  too 
largely  jdiysical  exercise  and  too  little  arithmetic.  In  playing  games 
keej)  the  groups  small  or  the  attention  is  scattered,  and  there  is  lack 
of  concentration.  The  rest  of  the  pupils  may  be  given  seat  work  re- 
lating to  some  other  subject  or  to  the  number  games  they  have  just 
jilayed. 


21 

The  child ren  should  keep  their  own  scores,  uo  matter  how  long- 
they  take  to  do  it.  (4ive  the  necessary  help  where  it  is  neede«l. 
Later,  a  child's  score  may  l>e  disrej^arded  if  he  cannot  yet  it  in- 
dependently. 

SUBJECT  MATTER  AND  PROJECTS. 

NOTATION  AND  NUMERATION.  Counting  hy  I's  hack  from 
fifty;  by  lU's  to  lUU,  beginning  at  U,  1,  2,  8,  etc;  by  o's  to  lUO;  by  2\s 
to50;  by  Vs  to  100.     lieading  and  writing  numbers  to  100. 

ADDITION.  The  addition  combinations  to  be  taught  are  as  follows: 
Group  I         000  0  000  0  000 
Very  Easy     0  1   2  8  4  5  6  7  S  9  0 

and  their  reverses. 
Group'TI       111111112  2  8  4  5 
Easy  2  8  4  5  6  7  8  9  3  2  8  4  5 

and  their  reverses. 
Arranged  by  Courtis  according  to  dificulty. 
Use  these  combinations  in  columns  which  involve  no  carry- 
ing. 
Teach  the  recognition  of  +  and  =. 

SUBTRACTION.  Involving  the  addition  combinations  taught  in 
this  gra<le.    Teach  the  use  of  the  sign  — . 

MEASURES.  Quarter,  half-dollar,  gallon,  pound,  dozen,  half- 
dozen,  house"  number,  the  date. 

Optional  Work. 

lieading  and  writing  numbers  to  150, 
Heading  and  writing  Roman  numerals  to  XII. 
Counting  by  2's  to  100. 
Table  of  2's. 

GENERAL  PROJECT. 

This  is  a  suggested  project  which  will  involve  much  of  the  sub- 
ject matter  of  the  grade.  Lest  it  should  not  provide  sufficient  drill 
to  fix  the  processes,  minor  projects  are  given  as  details  un<ler  each 
topic  of  sul)ject  matter.  The  latter  are  unrelated  but  might  be  an 
outgrowth  of  a  larger  project  similar  to  the  one  here  given. 

PLAYING  STORE.  For  this  project  each  child  may  make  liis  own 
store  as  an  Industrial  Art  problem.  Interest  in  the  project  may  be 
increased  by  having  different  kinds  of  stores  such  as,  a  grocery, dairy, 
toy  shop,  bakery,  etc^  Toy  money,  articles  of  nierchaudi.»<e,  casji  reg- 
isters and  safes  for  these  stores  may  be  made  during  the  Industrial 
Art  periods  and  seat  i)eriods. 

I>y  utilizing  this  ])roject.  direct  apjilication  of  the  various  topics 
in  the  II-I>  outline  may  be  made  as  follows: 

COUNTING  TO  100.  ChiMren  may  count  pennies  in  tlic  cash 
register.      See  that  each  store  keei>er  is  provided  with  at  least  lUd. 

Children  may  count  eggs,  balls,  marbles,  cookies  and  otlier  arti- 
cles of  merchandise  in  stock.  Counting  should  be  done  by  I's,  2's, 
aud  5's.     Storekeeper  may  have  a  clerk  to  check  up  with  him. 


READING  AND  WRITING  NUMBERS  TO  100.  Each  child  mayuum- 
l»er  his  store,  chtiusiuy:  a  munl'er  between  oU  to  lUU.  No  two  iiuinbers 
i<houM  he  alike.  These  uuinV)ers  may  be  written  on  cards  and  placed 
4>n  front  of  stores.  The  children  will  like  to  take  turns  in  reading 
the  store  numbers,  usinj;  comidete  sentences  as:  ''The  number  of 
.lohn's  store  is  9'i." 

Price  taifs  may  be  iiiaile  and  attached  to  artit-les  of  merchandise. 
Children  shouhl  be  eucourageil  to  tind  out  correct  jtrices  of  articles  in 
their  stores. 

Keadin":  price  tajjs  will  be  of  far  more  interest  than  just  reading 
numbers  written  on  the  board.  Complete  sentences  should  be  used, 
e.  jr.,  "One  poun<l  of  butter  costs  fifty-five  cents," 

MEASURES.  Quarter,  half-dollar.  During-  the  process  of  making 
his  toy  money,  the  child  rai)idly  learns  to  recognize  the  fliffereut 
pieces.  Customers  may  ask  storekeepers  to  change  a  dime  into 
nickels  or  pennies,  or  a  half-dollar  into  (piarters. 

l*t>und.  To  develoi)  the  concejtt  of  a  pound,  butter  cartons  filled 
with  sand  or  pound  candy  boxes  may  be  used.  Children  must  be 
taught  what  objects  are  sold  by  the  pound.  Encourage  criticism  of 
such  a  statement  as,  "I  want  a  pound  of  vinegar." 

(Gallon.  A  gallon  oil  can  in  the  grocery  store  and  a  milk  can  in 
the  dairy  will  give  a  correct  concept.  Children  should  learn  what 
l»roducts  are  sold  Ijy  the  gallon. 

Dozen  and  a  half-dozen.  After  the  class  has  been  taught  that 
any  twelve  articles  makes  a  dozen,  a  "dozen  lesson"  at  the  different 
stores  will  be  greatly  enjoyed.  Everything  must  be  purchased  by 
the  dozen  or  half-dozen.  Cutomers  must  count  over  the  articles  to 
make  sure  that  the  dealers  have  given  the  correct  number.  If  a  cus- 
tomer should  ask  for  one  dozen  sugar  or  tea  the  storekeeper  must 
correct  the  error  or  forfeit  his  position. 

HOUSE  NUMBER  AND  DATE.  Each  storekeeper  may  keep  order 
books.  On  each  order  slij)  he  may  write  the  name  and  address  of 
each  customer  and  the  date. 

Combinations. 

ADDITION.  Addition  combinations  will  necessarily  be  used  when 
two  articles  are  purchased  e.  g.,  a  pencil  for  Ic  and  a  ruler  for  5c. 
(5  +  1).  When  drill  on  a  i)articular  combination  is  required  the  chil- 
dren may  use  the  iiumbers  of  that  combination  in  making  their  pur- 
chases. The  storekeeper  may  use  the  combinations  in  checking  up 
his  stock,  e.g.,  5  cans  of  corn  on  one  shelf  plus  5  cans  on  another=10. 
Two  white  V)alls  +8  red  balls  =  5  balls. 

SUBTRACTION.  Since  the  Austrian  Method  is  to  be  taught,  the 
store  aftords  splendid  opi>ortunity  for  making  jiractical  apjilication. 
Making  «diange  will  fix  su])tracti(m  combinations  both  in  the  mind  of 
the  storekcepei-  and  the  customer.  i)rovideil  the  same  numbers  are 
used  frei)uently. 

Storekeepers  should  be  encouraged  to  check  up  sales  from  time 
to  time  as,  "There  were  ten  loaves  of  bread  on  the  counter.  Five 
loaves  are  left.  Tlit  re  were  five  loaves  soM.  (Five  and  what  number 
makes  ten  ?) 


•28 
Correlation  With  Other  Subjects. 

LANGUAGE.  Insist  upon  fiooil  Kn<ili,sli  from  the  rliiMii-n  who  are 
l)layint>-  store.  Encourafje  use  of  conijilete  sentences.  IJoth  ciistoiiier 
and  dealer  should  criticise  freely  any  errors  in  Enyflisli. 

Occasionally  let  the  children  tell  about  i>layin{j  st<jre  for  a  lan- 
iiuaye  lesson.    This  may  be  in  the  form  of  a  conversation  lesson. 

SPELLING.  Storekeepers  may  wish  to  write  labels  for  some  of 
their  merchandise.  A  few  simple  words  selected  from  the  store  vo- 
cabulary will  motivate  the  spelling  lessons.  The  words  learned  (lur- 
ing the  spelling  period  may  then  be  written  on  labels  and  pasted  on 
to  the  wares. 

REMARKS.  Occasionally  some  store  may  liold  a  sale.  Ked  letter 
price  tags  may  be  attached  to  articles  on  sale. 

As  a  reward  for  good  work,  some  child  may  be  made  an  inspector 
whose  duty  it  is  to  see  that  all  stores  are  in  proper  order  and  that 
price  tags  are  correct. 

Ordering  by  telephone  will  add  interest  to  the  store  ami  keep 
the  play  from  becoming  monotonous. 

MINOR  PROJECTS. 

NOTATION.  To  climb  up  and  down  the  ladder.  The  ladder  may 
reach  from  1  to  50  or  100  and  steps  may  be  Ts,  '2's,  o's  and  lO's. 

To  write  the  names  of  the  steps  on  the  ladder  and  then  read  tlie 
names. 

To  guess  who  is  counting.  A  child  comes  to  the  front  of  the 
room  and  hides  his  face.  A  child  i)ointed  out  by  the  teacher  counts 
by  2's  to  24.     The  one  in  front  guesses  the  narae.x 

ADDITION.  To  play  the  Family  Game.  Each  child  is  given  any 
niimber  from  I-IO.  A  number  such  as  18,  14,  lo,  16,  17,  IS  is  writ- 
ten on  the  board.  If  14  is  chosen  the  child  having  number  <>  finds 
number  9  and  stands  by  him.  All  the  combinations  are  thus  paire<l 
off.  The  cards  are  redistributed  and  the  same  or  a  different  number 
is  jdaced  on  the  board. x 

To  catch  the  most  fish.  Combination  cards,  each  representing  a 
fish,  are  on  the  fioor  or  on  benches.  The  child  catches  one  each  time 
he  calls  the  combination  correctly.  If  his  answer  is  wrong,  the  fish 
gets  back  into  the  water. 

To  get  an  ajjple.  An  apjtle  i^;  drawn  on  the  board.  <  )n  each  side 
of  it  are  written  combinations.  Two  chiltlren  now  lace  in  calling 
these  combinations,  the  first  one  through,  getting  the  apple. 

To  play  bean  bags.  Three  circles  of  different  sizes,  each  liaving 
a  different  value  as  2,  3,  or  4,  are  drawn  on  the  floor.  Two  leadeis 
choose  the  people  for  their  sides.  Each  cliild  throws  two  bags.  The 
score  for  each  side  is  determined  when  all  have  had  a  turn.  Adaptt-d 
from  the  Baltimore  Course  of  Study. 

To  climb  a  ladder.  Combinations  are  written  on  eacii  rung.  The 
game  is  to  see  who  can  climb  the  highest.  If  a  second  trial  is  given, 
the  child  i-an  see  how  many  rungs  higher  he  climlu-d  than  in  the  tirst 
trial. 


24 

tin. 

1  how  iiuK-h  thinjr!* 

w 

ill  cost 

1. 

A  picture  jtostcar 

.1 

Ic 

A  ruler 

3c 

•  ) 

A  liox  ot"  I'lialk 

Ic 

A  penlwtMer 

«c 

3. 

A  |iai)er  doll 

'Ic 

A  l.all 

t  c 

4. 

A  jitick  of  caudy 

8  c 

A  peueil 

8c 

Oral  work  oulv. 


The  child  answers  in  his  turn  as  follows:"!  want  to  buy  a  picture 
postcanl  f«>r  Ic  and  a  ruler  for  3c.     I  shall  pay  4c  for  both  of  them." 
X  Adapted  from  Decatur  Course  of  Study, 

SUBTRACTION.  The  first  four  projects  above  may  be  used  for 
subtraction  also. 

To  play  ]\Iore  or  Less.  .V  child  has  a  certain  number  of  objects 
concealed  in  his  hand.  He  asks, "How  mauy?"  Ans.  Five.  If  the 
answer  is  wrong-  he  says,"No,  it  is  two  more  than  five,"  or  No,  it  is 
three  less  than  five."  Adapted  from  the  Baltimore  Course  of  Study. 
To  <letermine  the  differences  in  prices:  How  much  more  does  a 
penholder  at  t)c  cost  than  a  l)ox  of  chalk  at  Ic?  A  ball  than  a  paper 
doll? 

To  find  out  how  much  more  I  must  save  to  buy  a  tablet,  a  pencil, 
crayons,  and  other  school  necessities. 

The  child  pretends  he  has  saved  Ic.     How  much  more  must  he 
save  to  buy  these  articles: 
A  paper  doll  at  2c 
A  i>encil  at  4c 
A  box  of  crayons  at  8c 
A  tablet  at  7c 
Child  answers:  '"I  have  saved  Ic,  and  a  tablet  costs  7c,  so  I  must  save 
♦ic  more  to  l)uy  it,"  etc. 

MEASURES.  To  play  store.  See  the  general  project  under  ad- 
dition. 

To  i>lay  milkman.  Let  one  child  be  milkman  and  come  to  each 
child's  house  (a  corner)  and  sell  him  a  i>int,  quart,  or  gallon  of  milk. 
This  may  involve  money  also. 

To  order  groceries.  Let  one  child  play  telephoning  to  the  store 
t<j  order  a  pound  of  butter  or  a  dozen  or  half-dozen  eggs,  giving  the 
house  numlter  to  the  clerk. 

To  invite  some  friends  to  a  Christmas  tree  party.  The  child  who 
is  inviting  must  give  the  house  number  or  the  guests  will  be  lost.  He 
may  also  give  the  date.  Note:  Do  not  ask  for  both  of  these  facts  at 
the  same  time. 

To  )»uy  thrift  stami)s.  Children  jday  at  buying  and  actually  buy 
one,  two  or  more  thrift  stami)S,  naming  the  coins  they  use. 

To  give  and  receive  correct  measure.  One  child  is  the  grocer, 
while  others  are  customers  leaving  their  orders.  Pint  and  quart  milk 
bottles  are  used  to  show  liquid  measures,  while  cardboard  objects 
serve  for  use  in  counting  out   the  dozen    and    half-dozen.     The  child 


25 

should  he  eucouratred  to  ask  the  yrocer  for  many  different  thinjjf*  s<»l<I 
hy  each  Tneasure,  thus  faniiliarizinfj  himself  with  the  use  of  tliese 
measures. 

Standards  of  Attainment. 

See  Standards  of  (yrade  II-A. 

Bibliography. 

See  (irade  II-A. 


•2(3 

(iKAl)K   II-A. 

DIRECTIONS. 

DUISION  OF  TIME.  Oue-half  of  the  iirithmetic  time  allotment 
in  this  trraile  should  be  giveu  to  oral  work. 

One-half  of  the  work  should   relate  to  concrete  problems. 

RE\1H\V.  Review  measures  taught  iu  Grade  II-B,  counting  to 
100  by  Ts  and  ")\s,  to  50  by  2's;  backward  from  50  by  Ts.  Column 
addition  involving  the  very  easy,  easy  and  medium  difficult  combina- 
tions. 

MAIN  TOPICS.    Addition  and  subtraction. 

Head  the  general  directions  for  (4rade  II-B.  Apply  the  sug- 
gestions given  there  for  counting,  reading  and  writing  numbers  and 
work  on  combinations  in  this  grade  also. 

ADDITION.  ''Before  extended  drill  in  addition  is  begun,  entire 
familiarity  with  the  number  scale  to  200  must  be  obtained  as  shown 
}»y  ability: 

To  read  the  numl)ers. 

To  write  the  numbers. 

To  visualize  the  number  symbols. 

To  know  that  any  number  and  1  more  give  the  next  number  in 
the  scale,  and  that  the  number  that  immediately  precedes  any  number 
iu  the  scale  is  one  less  than  the  number,  and  that  one  less  than  any 
number  is  the  number  preceding  it. 

To  count  by  tens,  beginning  with  any  number. 

To  know  the  sum  of  a  given  decade  and  any  number  less  than 
10,  as:  20+5=25;  S0+6=86;  90+3=93,  etc. 

'Use  the  combinations  mastered  in  building  columns  for  adding. 
2  3  3  An  examination  of  these  illustrative  columns  will  show  how 
6     5     4     columns  can  be  constructed  and  extended  upward  as  far  as 

2  2     3     the  teacher  likes. 

3  9     3 

4  3     4 
3     6     5 

3  2     6 

4  3     2 
3     4     9 

2  5     3 

In   l>eginning  column    addition    with    children    in    the    primary 

3  grades.  ]tlace  the  following  column    on  the  l)oard.     Take  the 

4  clialk,  and,  beginning  at  the  foot  of  the  column,  say:  "Two, 
3  three,  five,"  ijointing  to  the  numbers  as  named,  and  write  the 
9  32  5  to  the  right  of  the  3.  Then  say,  "Five,  four,-nine."  Write 
3  23  the  9  to  the  right  of  4.  Then  say,  "Nine,  three.-twelve,"  and 
6  20     write  12  to  the  right  of  the  3.    Then  continue,  '  Twelve,  two,- 

2  14     fourteen,  writing  the  14  to  the  right  of  the  2,  and  so  on  until 

3  12     the  column  is  added.    At  each  step  have  the  ehihlren,  collect- 

4  9  ively  or  imlividually.  repeat  after  you  each  statement.  Drill 
3  5  the  pupils  until  they  can  go  through  this  without  error.  If 
2 there  is  any  hesitancy  about  the  combinations,  point  to  the 


27 

combination  above,  so  that  they  may  learn   where  to  find  the 
correct  form  if  they  shouM  fory-et. 

After  this  process  and  language  form  is  established,  write  similar 
columns  on  the  board  for  each  pupil,  with  instructions  for  him  to  do 
the  exercise  himself.  The  teacher  should  pass  from  one  to  anotlier, 
hearing  each  give  the  form.  As  a  pu])il  tinishes,  let  liim  exchange 
examples  with  another  pupil,  first  erasing  the  side  columns.  To  a- 
void  confusion,  it  is  well  to  write  two  or  three  examples  in  excess  of 
the  number  in  the  class,  so  that  no  pupil  need  wait.  As  a  furtlier 
convenience,  it  may  be  helpful  for  the  pupil  who  iinislies  a  column  to 
write  his  name  underneath  it.  The  teacher,  passing  around,  later 
erases  the  answer  and  the  side  columns,  and  writes  "C"  (correct)  or 
"X"  (wrong)  after  his  name.  The  place  is  then  ready  for  another 
pupil. 

With  a  few  pupils,  there  will  be  a  continual  tendency  to  make 
mistakes  in  the  left-hand  figure,  to  write  42  instead  of  32,  etc.  This 
means  that  insufficient  work  has  been  done  on  the  number  scale. 
Suppose,  as  in  the  illustration  given,  the  pupil  writes  42  instead  of 
32.  To  correct  this,  several  methods  are  at  the  option  of  the  teacher, 
(l)  She  can  go  back  for  more  drill  in  the  decades,  then  make 
the  application  to  the  difficulty  in  hand"  (2)  8he  may  have 
him  write,  in  ascending  column,  the  number  beginning  with 
23,  until  the  next  2  is  reached.  (3)  She  may  draw  a  line  un- 
der 23,  and  ask,  "What  2  next  above  23?"  (Answer,  "32.") 
After  the  combinations  already  mentioned  have  been 
mastered,  and  every  child  can  work  out  the  side  columns  of 
any  column  of  figures  built  up  out  of  these  combinations, 
readily  and  without  mistake,  the  same  combinations,  in  their 
reverse  form,  should  be  treated  in  like  manner. 


The  purpose  of  this  is  to  drill  the  pupils  in  learning  new  com- 
binations and  in  visualizing  the  end  figures  of  the  successive  partial 
sums.  After  this  form  has  been  mastered,  the  teacher  should  con- 
tinue ad<lition  without  writing  the  sums  at  the  side,  and  train  the 
pujiil  to  add  without  this  help.  In  starting  this,  it  is  well  to  re<iuire 
the  pupil  to  add  directly,  thus:  five,  nine,  twelve,  fourteen,  twenty, 
twenty-three,  etc.  If  he  makes  mistakes,  have  the  pupil,  in  imagina- 
tion, go  through  the  form  of  the  partial  sums  in  the  side  column, 
without  actually  writing  them.  First  attemi>ts  will  be  slow,  but  a 
few  exercises  will  cause  him  to  depend  ui)on  his  own  visual  imagin- 
ing.    Proceed  in  the  same  way  to  add  other  columns  in  review. 

In  all  this  early  work,  the  child  should  never  be  permitted  to 
perform  any  work  in  addition  at  his  seat,  V)ut  always  at  the  board,  in 
full  view  of  the  teacher.  Children,  if  allowed  the  time,  will  fall  back 
into  the  habit  of  counting  up  the  sums  serially.  It  is  a  mistake  to 
think  that  chihlren  will  outgrow  this  habit,  once  it  is  formed.  Chang- 
ing one's  habits  is  not  so  simple  a  matter  as  this.  To  jirevent  this 
habit  from  being  formed,  the  teacher  must  first  give  in  columns  only 
those  combinations  which  the  children    have  first  learned  thorougiily. 


3 

4 

3 

9 

42 

3 

23 

6 

20 

2 

14 

3 

12 

4 

9 

3 

5 

2 

28 

au«l,  sec'oud,  always  iusist  that  the  work  he  performed  at  the  V^oard 
ami  ill  full  view  of  the  teacher.  Do  uot  permit  the  child  to  stop  and 
think.  He  either  knows  the  sum  or  uot.  If  he  shows  the  least  hesi- 
tancy he  must  either  be  told  the  answer  or  l)e  permitted  to  look  at 
the  i-omhiuation  involved  in  the  answer.  For  this  purpose  the  com- 
hinatioiis  should,  with  their  sums,  always  lie  written  on  the  hoard  in 
full  view  of  the  child. 

Concert  work  is  y:ood,  hut  it  should  not  be  eiiii)l()yed  exclusively, 
for  many  children  are  thereby  made  dependent  in  their  work.  Again, 
if  a  teacher  uses  it  too  generously,  she  cannot  know  what  the  indi- 
vi<luals  are  capable  of  doing.  In  addition  work,  the  teacher  must 
keep  in  mind  the  fact  that  her  class  will  not  pi'oeeed  uniformly  in 
the  acquisition  of  the  work,  and  that  in  consequence  she  must  provide 
some  way  to  give  much  individual  instruction."  Los  Angeles  Course 
of  Study. 

SUBTRACTION.  See  Grade  II-B  which  gives  suggestions  relating 
to  the  Austrian  Method  of  Subtraction. 

MULTIPLICATION,  DIVISION  AND  FRACTIONS.  Introduce 
multijdication  as  a  short  method  of  addition.  If  a  child  through  fre- 
(juent  repetitions  knows  that  8+8+8  ai'e  always  9,  the  transition  to 
three  8"'s  are  9  will  V)e  easy.  Teach  a  few  multiplication  com))inations 
at  a  time  rather  than  the  whole  table.  Each  combination  and  its  reverse 
should  be  taught  together,  as  two  S's  are  10  and  five  'i's  are  10.  The 
f(»rm  for  the  tables  'i-'i's  are  4,  8-'2's  are  6,  4-2\s  are  S  leads  the  child . 
easily  toward  division.  If  he  knows  that  8-2's  are  6,  he  can  readily 
answer  the  question,  "How  many  2''s  are  there  in  6?"  orally.  Division 
and  fractions  should  be  used  with  multiplication.  Thus  one  combina- 
tion may  take  six  different  forms: 

Five  2's=10. 

Two  5^8=10. 

10--2=5. 

10-5=2. 

\  of  10=5. 

I  of  10=2. 

Variety  in  expression  deepens  the  general  impression  of  the  rela- 
tive values  of  2,  5  and  10. 

PROBLEM  WORK.  All  problems  are  to  be  oral  and  based  on  the 
child's  experience.  Present  only  one  step  problems  with  simple 
numl>er8. 


SUBJECT  MATTER  AND  PROJECTS. 

NOTATION  AND  NUMERATION. 

Iteading  and  writing  numbers  to  200. 
Rea<ling  and  writing  Roman  numerals  to  XII. 
Counting  to  4S  by  4^8  and  6'8, 
Counting  to  80  bv  8'8. 


I 


'2 

3 

8 

:} 

8 

8 

4 

2 

(> 

2 

4 

5 

() 

7 

5 

7 

8 

4 

4 

4 

♦) 

1 

S 

\) 

7 

1 

S 

9 

«) 

< 

S 

» 

29 

ADDITION. 

Addition  ( 'ombinatious. 

Group  III  2       2       2 

Average  8'      4       o 

Group    IV  2       2       8 

Hard  S       9       8 

Arrauged  by  Gourtis. 

SUBTRACTION.  Subtraction  combinations,  the  reverse  of  the 
addition  combinations, 

MULTIPLICATION.    Construct  and  learn  tables  of  2\s,  TVs  and  lO's. 
Teach  the  symbol  X  as  "times." 

1X5=5 
Use  this  form  for  the  tables:     2X5=10 

3X5=15 

DIVISION.     Teach  the  reverse  of  the  multiplication  tables. 
Teach  the  symbol  ~^, 

DENOMINATE  NUMBERS.  Names  and  number  of  the  days  of  the 
week. 

Names  and  number  of  the  months  in  the  year. 
The  hours  on  the  clockface. 
The  signs  §  and  c. 

CONCRETE  PROBLEMS.    Problems  relating  to  the  following: 
The  Milkman. 
Carfare. 
Marketing. 
A  Bird  Calender. 
The  School  Week  and  Month. 

FRACTIONS.     Fart-taking: 

2  of  the  numl)ers  to  24  which  give  an  integer  as  a  result. 
.'  of  the  numbers  to  50  which  give  an  integer  as  a  result. 
u)  of  the  numbers  to  50,  which  are  exactly  divisible. 

OPTIONAL  WORK. 

Extensive  work  in  addition  an<l  subtraction.     Caution:  Keep  tin- 
work  simple  enough  to  prevent  counting. 
TaV)le  of  8\s. 

GENERAL  PROJECT. 

This  is  a  suggested  project  which  will  invi»lve  nnu-li  of  tiu-  sub- 
ject matter  of  the  gra<le.  Lest  it  should  not  provide  sutlicietit  drill 
to  ftix  the  processes,  minor  projects  are  given  as  tletails  under  each 
topic  of  subject  matter.  The  latter  are  unrelate<l  but  might  be  an 
outgrowth  of  a  larger  jtroject  similar  to  the  one  here  given. 

Playing  Cafeteria.  Individual  cafeterias  for  this  proj«'<-t  may 
be  constructed  out  of  wooden  or  cardboard  boxes,  during  the  Industrial 
Arts  period.      If  desired,  one  large  cafeteria  maybe  rniistni.t..,)    .••ich 


30     - 

child  haviutr  some  part  in  the  txmstnu-tion  or  furuishing.  Toy  money, 
toy  foods,  cash  retristers  and  <li.slies  may  l)e  maile  during-  Industrial 
Arts  or  seat  i)eri(.)ds. 

The  various  topics  ou  II- A  outline  may  be  introduced  into  this 
project  as   follows: 

READING  AND  WRITING  TO  NUMBERS  200.  The  cafeterias  may 
l)e  numbered,  only  uuml)ers  between  lUO  and  '200  being-  used.  The 
children  may  then  take  turns  in  readin":  the  numbers  on  the  different 
cafeterias. 

READING  AND  WRITING  ROMAN  NUMERALS  TO  XII.  Each 
cafeteria  will  need  a  clock.  The  proprietors  may  make  these  clocks 
and  set  them.  Suggest  to  the  children  that  they  talk  about  the  time 
occasionally,  e.  g..  "Why,  it's  nine  o'clock  already." 

COUNTING.  By  frequently  counting  the  money  in  their  registers 
])y  B's,  4's,  and  6's  the  children  may  obtain  skill  in  counting. 

MULTIPLICATION.  Application  of  tables  may  be  made  in  figur- 
ing costs;  e.  g.,  If  one  cookie  costs  B  cents  what  must  a  customer  pay 
for  five?  If  one  potato  costs  4  cents  what  must  a  customer  pay  for 
t  wo? 

DIXISION.  Playing  cafeteria  also  involves  knowledge  of  the  use 
of  division,  e.  g.,  a  proprietor  may  have  20  sandwiches  ready  to 
serve.     If  each  customer  buys  two,  how  many  may  be  served? 

A  patron  may  have  12c  left  for  cakes.  How  many  cakes  can  he 
purchase  if  they  are  priced  3  cents? 

ADDITION.  The  customer  need  not  pay  for  his  order  until  he 
receives  a  check  on  which  all  the  items  have  been  adde<I. 

At  the  end  of  the  day  the  proprietor  may  add  up  all  his  sales 
and  place  the  sum  on  his  file. 

SUBTRACTION.  Drill  in  suljtractiou  combinations  may  be  ob- 
tained by  making  change. 

DENOMINATE  NUMBERS.  In  making  bills  of  fare  the  children 
will  soon  become  familiar  with  the  8  and  c. 

If  each  cafeteria  will  have  a  day  calendar  there  will  be  little 
difficulty  in  learning  the  names  of  the  months  an<l  days. 

CONCRETE  PROBLE.MS.  Following  a  lesson  at  the  cafeteria,  the 
children  should  ]>e  encouraged  to  make  up  problems  about  their  pur-j 
chases.     Complete  statements  should  be  required  in  the  solutions. 

Correlation. 

L.\NGl  .\(  .E.  C'omijlete  statements  shouhl  be  required  at  all  times. 
Any  errors  in  grammar  should  be  corrected  immediately  by  the  chil- 
dren themselves. 

Occasionally  the  class  may  write  short  sentences  about  their  cafe- 
teria. 

A  .><tu<ly  of  the  different  foods  served  will  jtrovide  language  les- 
sons that  are  well  woith  while. 


81 

SPELLING.  The  chiMren  may  want  to  know  how  to  spell  differ- 
ent words  for  their  l)ills  of  fare.  A  few  of  these  words  may  he  intro- 
duced into  the  spelling  lessons. 

IXDUSTRL\L  ARTS.  Bills  of  fare,  i)osters,  toy  food,  money,  rejjiw- 
ters,  and  dishes  may  be  made  in  the  Industrial  Arts  class.  Different 
mediums  may  be  emj)loyed,  such  as  clay  ni(idellin<r.  i)aper  cuttiutr, 
construction  work,  etc. 

INSTRUCTIONS. 

Prices  on  bills  of  fare  must  be  correct.  Chihlreu  should  find  out 
the  prices  for  themselves. 

Bills  of  fare  should  be  changed  frequently. 

A  child  who  makes  au  eror  in  Arithmetic  while  acting  as  cashier 
or  proi)rietor  must  forfeit  his  position. 

MINOR  PROJECTS. 

NOTATION. 

To  play  "Buzz."  The  children  count  by  3's,  4's,  6's,  substituting 
the  word  "Buzz"  for  the  number  chosen  as  1,  2  Buzz  -4,  5  Buzz,  etc.  x 

ADDITION  AND  SUBTRACTION. 

To  guess  numbers.    A  child  stands  before  the  class  saying: 
'T  am  thinking  of  two  numbers  whose  sum  is  I'i." 
"Are  they  6  and  6  ?" 
||Xo." 

"Are  they  5  and  7  V" 
les. 

The  pupil  guessing  correctly  goes  to  the  front,  x 

To  match  sums.  The  teacher  or  a  child  holds  a  card  with  a  num- 
ber combination  as  6+6.  The  one  who  can  think  of  a  number  com- 
bination whose  sum  is  the  same  as  the  teacher's  stands.  If  wrong,  he 
must  sit. 

To  play  Tenpins.  Each  teni)in  is  given  a  detinite  value.  A  child's 
score  is  the  total  value  of  all  the  teni)ius  he  knocks  down  in  one 
turn.  X 

To  play  Cat  and  ,AIouse.  A  circle  is  drawn  on  the  hoard,  num- 
bers being  written  on  its  circumference.  A  number  is  placed  in  the 
circle  to  be  combined  with  the  numbers  around  it.  One  child  is  the 
cat.  He  points  quickly  to  a  number  and  calls  on  a  classmate.  The 
I)upil  who  fail  is  caught  and  put  out  of  the  game.  If  the  cat  does 
not  notice  a  mistake,  one  of  the  mice  takes  his  place,  x 

To  play  Dog  and  Lost  Sheep.  Each  child  has  a  combination  of 
any  one  number  as  10.  One,  the  dog  has  nothing.  The  ilog  says.  I 
am  looking  for  a  lost  sheep.  Are  you  5-|-5  V"  "No,  I  am  not  10." 
The  dog  goes  to  another  sheep  saying  "Are  you  4-+-6':'"  and  so  on 
until  he  guesses  the  right  one  who  answers,  "Ves,  I  am  'i-fS."  The 
dog  then  chases  the  shec])  until  he  is  caught,  and  the  sheep  becomes 
the  tlog.  X 

To  avoid  being  put  oft  a  street  car.  The  children  are  in  two  rows 
facing  each  other,  each  having  a  combination  card.    One  dnld  is  c«)n- 


32 

.liK-tor;  child reu  pay  him  their  fare,  which  is  the  answer  to  the  com- 
t)iuatiou.  If  the  coinbiuatiou  is  uot  correctly  called,  the  child  is  not 
aide  to  pay  his  fare  aud  is  put  off  the  car.  The  children  able  to  re- 
main on  the  car  the  longest  are  the  winners. 

To  tret  the  most  valentines.  Cards  bearinjj  combinations  (valen- 
tines) are  put  int<»  valentine  box  havinjr  a  bigoi)ening  in  the  cover. 
The  children  then  pull  out  valentines,  keeping-  them  if  the  answers 
to  their  coinbinations  are  correctly  calle<l. 

To  tin<i  from  the  jtrice  list  how  much  some  candy  and  a  tup  will 
cost;  a  whistle  and  a  toy  pistol:  dominoes  and  an  orange;  a  ball  an<l 
a  l>ook. 

To  tind  how  much  more  a  pistol  costs  than  an  orange;  a  paper  pad 
than  a  top:  a  truin]n't   than  a  whistle. 

rinOK  list: 

Dominoes 9c  Top 4c 

Whistle tic  ]>ook 9c 

Trumpet 8c  Ball 7c 

Candy 7c  Orange 5c 

Toy  pistol 8c  Apple 6c 

I'ajter  pad 5c 

X  Adajiteil  from  the  Decatur  Course  of  Study. 

MULTIPLICATION  AND  DIVISION.  To  climb  a  ladder.  Two 
ladders  each  containing  combinations  are  placed  on  the  board.  Pupils 
race  up  and  down  the  rows,  or  see  how  high  they  can  climb. 

To  sjtin  the  arrow.  A  circle  of  cardboard  in  the  center  of  which 
is  fasteneil  an  arrow,  has  numbers  on  the  circumference.  A  child 
spins  the  arrow,  an<l  the  class  gives  the  i)ro<luct  of  the  number  on  the 
arrow  and  that  on  the  circumference.  The  pupil  who  gives  the  an- 
swer first  may  spin  the  arrow  in  the  next  game. 

To  race.  Products  are  written  upon  the  board.  Two  chiltlren 
are  chosen  to  see  who  can  most  quickly  point  to  the  answer  to  a  com- 
bination given  by  the  other  pupils.     Scores  may  be  kept. 

To  see  who  can  get  the  most  cards.  Cards  containing  combina- 
tions are  jtlaced  along  the  board.  Two  children,  starting  at  either  end 
race  to  see  who  can  reach  the  center  first. 

To  i»lay  (irab.  P^ach  child  in  turn  grabs  a  card  from  the  (rrab 
I5ag.  If  he  gives  the  correct  answer,  he  may  keep  the  card;  if  he  gives 
the  wrong  answer  he  must  i)ut  the  card  back.  The  one  holding  the 
most  cards  at  the  end  is  the  winner.      The  Detroit   Course  of  Study. 

To  plan  for  a  party.  Mother  has  15  cakes,  20  sandwiches,  and 
10  pieces  of  candy.     There  will  be  five  girls  at  the  party. 

How  many  cakes  will  each  get?  How  many  sandwiches?  Pieces 
of  caiiily? 

I'se  this  same  idea  for  multiplication. 

To  find  out  how  much  it  will  cost  to  go  downtown.  If  there  are 
five  boys,  what  will  it  cost  to  go  downtown?    Three  boys?    Seven  girls? 

DENOMINATK  NCMBERS. 

(Merely  SugK-eHted) 

To  make  a  monthly  bird  calendar,  putting  in  the  names  of  the 
days  of  the  week. 


33 

To  find  various  thin<>-s  on  the  calendar.  The  number  of  days  in 
a  week.  The  number  of  Sundays  or  of  any  other  day.  Tlie  date  on 
which  the  second  Monday  occurred,  etc.  The  day  on  wliii  li  tin-  four- 
teenth, tenth,  etc.,  occurred. 

To  make  a  clockface,  usiny  Koinan  numerals.  I'laciniif  the  hands 
of  the  clock  at  risinjif  time,  bey-i lining  school  time,  the  various  hours, 
Curfew  time.  Findint>-  how  many  hours  it  takes  the  hand  to  iro  from 
9  in  the  morning:  to  twelve  in  the  morniny;  from  2  in  the  afleriioon  to 
5  in  the  afternoon,  etc. 

To  write  the  number  of  <lollars  investe<l  in  the  school  Saviii<;s 
Bank.     To  find  the  total  number  of  dollars  saved  in  the  room. 

To  write  the  number  of  cents  spent  for  Thrift  8tami)s,  individu- 
ally and  as  a  class.  If  the  sum  is  larg^e,  the  teacher  may  <lo  the  add- 
intj,  beinfj  assisted  by  the  class. 

To  go  marketinjjf  usiny  toy  money.  Local  conditions  will  deter- 
mine the  j)articular  forms  this  may  take. 

Standards  of  Attainment. 

Woody  Test:  Series  A. 
Addition  6.8 

Sul)traction         5.1 

Bibliography. 

Brown  and  C'offman:   How  to  Teach  Arithmetic. 

Chap.  X  Primary  Arithmetic. 

Chap.  XI  Teachiu{>-  the  Fundamentals. 

Pp.  182-186  Part  Taking. 
Iloyt  and  Peet:   Everyday  Arithmetic,  JJook  I. 
Kendall  and  Mirick:  Teaching  the  Fundamental  Subjects. 

Pp.  170-190.  Skill  in  Calculation. 

Skill  in  Api)lication. 

Inductive  Teaching. 

Mental  and  Oral  Lessons. 

Klapper:  The  Teaching  of  Arithmetic. 

Pp.  49-51.   Motivation  in  Arithmetic. 

Pp.  70-75.  ()V)jective  Teaching. 

Pi*.  88-91.  The  Drill. 

Pp.  100-108.   Rationalization  of  Processes, 
Testing  Ability. 
Fundamental  ( )perations. 

Pj).  158-1(S2.   Addition  and  Subtraction. 

Pp.  182-208.  Multiplication  and  Division. 
Strayer  and  X'orsworthy:   How  to  teach. 

Pp.  204-205.  The  Drill   Lesson. 
Suzzalo:  The  Teaching  of  Primary  Aritliiiu't  ir. 

Chap.  IX  Si)ecial  Methods  for()l>taining  .Xcfuracy  and  Speed. 
Wilson:   Motivation  of  School  Work. 

Pp.  158-165.  The  Motivation  of  Aritliinetic. 


34 

GRADE  III-B 
DIRECTIONS. 

L)I\  ISION  OF  TIME.  Three-fourths  of  the  arithmetic  time  allot- 
meut  ill  this  trra<le  shouhl  be  fjiveu  to  oral  work.  Oue-fourth  of  the 
work  should  relate  to  concrete  problems. 

RE\IE\V.      Keadiiijr  and  writing-  numbers  to  200. 
The  addition  and  subtraction  combinations. 
Tables  of  ^'s,  S's,  and  lU's  both  in  multiplication  and  divis- 
ion. 

MAIN'  TOPICS.     Addition,  subtraction,  and  multiidication. 

Read  the  directions  for  Grade  II-A  relatiufj-  to  teaching  of  addi- 
tion. multii»licatiou,  and  division. 

.\DDITIO.\'.  Do  not  drill  on  combinations  already  mastered. 
Concentrate  attention  only  on  those  number  facts  and  processes  of 
which  the  children  are  not  sure.  This  will  require  that  the  teacher 
does  preliminary  and  constant  testing  to  discover  class  and  individual 
weaknesses. 

Hesitancy  in  giving  combinations  should  not  be  permitted;  for 
nothing  so  directly  promotes  guessing  or  counting  the  figures.  Instead, 
tell  the  child  the  combination  until  he  has  memorized  it.  A  combina- 
tion that  is  not  entirely  automatic  with  the  child  should  never  be  used 
in  a  concrete  problem.  Both  forms  of  combinations  in  addition  and 
multiplication  should  be  given.  Knowledge  of  3+9=12  does  not  im- 
l)ly  the  knowledge  of  9+3=12.  The  same  applies  to  3  -5  and  5X3. 
Provide  variety  in  work  on  the  combinations  and  tables. 

Teach  slowly  and  carefully  in  this  order  (l)  the  ])rocess,  (2)  drill, 
(3)  application.  The  more  eftective  the  teaching  the  less  need  of 
drill.  Avoid  teaching  by  drill.  Grouj^  work  for  those  needing  spec- 
ial help  should  be  organized,  but  ordinarily  the  teacher  shouhl  aim  to 
hold  the  attention  of  all.  The  problems  should  be  made  to  increase 
gra<lually  in  <lifticulty  by  increasing  the  number  of  orders  and  the 
uumVjer  of  addends. 

SUBTRACTION.  Use  the  Austrian  or  making  change  method  of 
subtraction.     For  a  discussion  of  this  method,  see  grade  II-B. 

MILTIPLICATIOX  AND  DIVISION.  The  child  should  not  be 
conij)elleil  to  give  a  reason  for  every  process,  for  he  learns  to  do  by 
doing.  Make  explanations  short  and  to  the  point,  giving  reasons  for 
the  processes  only  when  an  especial  value  in  doing  so  is  apparent. 

Addition,  subtraction,  multiplication  and  division  should  be  drill- 
ed ujMtn  simultaneously. 

liefjuire  the  children  to  say  ''the  difference  is";  "tlie  subtrahend 
is";  "the  product  is";  thus  familiarizing  them  with  the  language  in 
arithmetic  through  its  correct  use.  Be  sure  that  the  terms  multipli- 
cand, rnultiidier,  jiroduct,  divisor,  dividend,  and  quotient  are  under- 
8to(Ml  an<l   correctly  use<l. 

CONCRETE  PROBLEMS.       Pupils    work    on    concrete    problems   | 
should  in  this  grade  l)e  oral.     In  beginning  work  with   concrete   pro- 
blems,  have   all    i)roblems   in  one   lesson    relate  to   the  same  process.. 
At  another  tiirie.  jirobJems  relating  to  another  i>rocess  may  be  used. 


I 


35 
SUBJECT  MATTER  AND  PROJECTS. 

NOTATION  AND  NUMERATION.     Kea-liiiy-  an-l  u  litii.n  iiunilMTs 
to  5000. 

Komau  numerals  to  XX. 
Counting  by  2  and  4    to  100. 

ADDITION.    Combinations. 

Group  V 5     5     0     5     6     0     6     7     7     S 

Very  hard 6     7     8     9     7     8     9     8     0     0 

Two  and  three  place  columns  involving  carryiny-;   not  over  five 
addends,  as                                 34  345 

67  672 
82  823 
25  246 
14     325 


Also  mixtures  of  one, 

two  an 

3 

35 

372 

4 

316 

id  three  place  columns,  as 

Drill  on  endings,  as 

6         16 

26 

36         7         17 

9           9 

9 

9         9           9 

SUBTRACTION.    Easy  subtraction,  as 

146  235  397  295 

24  21  72  53 

Subtraction,  with  one  ste])  borrowing,  as 

82  54  9.S  192 

78  46  69  56 

Subtraction  of   two   i)lace  numbers   from   three   and    four   place 
numbers,  as 

4,S23                   .").3iMi 
612  22 

Work  on  endings,  as 

15  25  35  45         1(5         26 

_6  _6         _6         J>       __I        _i 

MULTIPLICATION.    Tables  of  3's  and  4's. 

Written  i)r(»blems  involving  the  one-jdace  multiplier  only. 


86 

DIXISloN.  Divisiuu  only  as  the  reverse  of  multiplicatiou  with 
no  carryiuti.     I'se  the  long  divisiuu  brac^. 

DENOMINATE  NL'MBERS.      The  ounce  and  yard. 
Time:  the  hour,  half-hour  an<l  quarter  hour. 

CONCRETE  PROBLEMS.      Problems  relating  to  these  subjects. 
Travel  and  In<lustry. 
Weitjhiug'  and  measuring  pupils, 
liuyinij  from  a  cataloicue. 
Siiarinji:  a  i)urchase. 
Time  tables. 

FR.ACTIONS.  Part-takinir  of  I  of  the  numbers  to  UU  and  4  of  the 
numV»er  to  SO  which  are  exactly  divisible. 

Addition  and  subtraction  of  very  simjde  fraction — no  reduction. 

OPTIO.VAL  WORK.      Practice  in  addition  and  subtraction  extend- 
ed as  far  as  possible  until  so  ditficult  that  counting  is  necessary. 
Multiplication  by  two-place  multipliers. 
Addition  and  subtraction  of  simple  fractions. 

GENERAL  PROJECT. 

This  is  a  suggested  project  which  will  involve  much  of  the  subject 
matter  of  the  grade.  Lest  it  should  not  provide  sufficient  drill  to  fix 
the  processes,  minor  projects  are  given.  The  latter  are  unrelated  but 
migiit  be  an  outgrowth  of  a  larger  project  similar  to  the  one  here 
given. 

Playing  Post  Office.  As  an  Industrial  Arts  problem,  the 
children  may  construct  a  toy  post  office  with  two  different  windows, 
one  for  stamps  and  another  for  parcels  post.  They  may  also  construct 
two  large  boxes,  one  for  mailing  letters,  the  other  for  packages. 

One  child  may  sell  stamps,  another  weigh  packages,  a  third  act 
as  sorter  and  a  fourth  as  postmaster.  The  post  office  should  contain 
as  many  small  letter  boxes  as  there  are  children  in  the  class. 

Either  toy  stamjis  or  cancelled  stamps  may  be  used  for  postage. 
The  children  may  make  their  own  envelopes.  If  a  pattern  is  used, 
enough  envelojtes  may  be  made  during  one  seat  period  for  several 
po.st  office  lessons.  Each  envelope  need  not  necessarily  contain  a 
letter.  JJooks  or  ])oxes  filled  with  sajid  may  be  used  for  packages.  A 
seat  j»eriod  may  be  utilized   for  coining  toy  money. 

The  following  are  suggestions  showing  how  the  subject  matter  of 
the  III-]>  Arithmetic  outline  may  l)e  applied  in  this  i)roject: 

READINC,  AND  WRITING  NUMBERS  TO  5000.  Each  post  box 
should  be  numliered.  If  the  children  will  use  only  numbers  between 
lOUU  and  ')(!(»()  they  will  soon  become  familiar  with'these. 

COUNTIN(;  HV  2  AND  4  TO  100.  The  stamp  seller  may  count  his 
pennies  by  *2'm  and  4's. 

The  sorter  may  count  letters  in  the  same  manner.  Daily  practice 
of  this  nature  tends  to  develop  swift  counting. 


37 

ADDITION.  The  staniii  seller  may  keejt  (•ouiit  of  the  iminl)er  of 
stanips  sold.  At  the  eii<J  </^^he  day  he  may  acM  the  total  nuiiil)er 
sold.    This  will   involve  volimie  ad<litioii. 

The  sorter  may  keep  a  daily  record  of  the  iiuiiiher  of  letters  de- 
livered. At  the  end  of  the  week  he  may  theii  find  the  sum  of  all 
letters  handled. 

The  child  who  Imys  stamps  must  also  ajtply  combinations  e.  ^.,  in 
purchasing  nine  Ic  stamps  and  four  '2c  stamps,  9+^  should  be  imme- 
diately recalled.  A  jturchase  of  16c  and  one  of  9c  means  drill  on 
endinifs. 

SUBTRACTION.  Makinir  change  at  the  stamp  ami  jiackaire  win- 
dows jjfives  practice  in  subtraction. 

MULTIPLICATION.  Practical  aj^plication  (.f  tables  is  ma-le  when 
the  child  purchases  four  2c  stamps,  eight  8c  stam]is.  etc. 

The  package  weigher,  who  uses  a  very  simple  table  of  postage 
rates,  may  be  asked  the  amount  of  postage  required  ou  a  8-pound 
package,  the  rate  being  5  cents  a  pound. 

DIVISION.  Division  comes  into  practical  use  when  a  child  con- 
fronts a  problem  of  this  nature:  I  have  18  cents;  how  many  2  cent 
stamps  can  I  buy? 

DENOM I  NATE    NUMBERS. 

Ounce.  If  the  class  is  particularly  advanced,  a  scale  showings 
ounces  as  well  as  i)ounds  may  be  used  for  weighing  packages. 

Time.  I'ost  office  employees  may  be  asked  questions  concerning 
the  time,  e.  g..  When  will  my  mail  l)e  ready?  When  does  this  office 
close?  The  time  may  be  shown  by  using  a  pasteboard  clock  with 
movable  hands. 

Suggested  Correlations. 

LANGUAGE.  Learning  to  write  letters  and  address  them  prop- 
erly is  a  Language  project.  The  Post  Office  project  will  motivate  the 
Language  lessons. 

Occasionally  the  children  may  write  a  short  paragraph  on  the 
Post  Office  (iarae. 

INDUSTRIAL  ARTS.  Most  of  the  materials  nee.ied  for  the  P(.si 
Office  may  be  made  during  the  Industrial  Arts  period. 

REMARKS.  Allow  children  to  take  turns  in  playing  s(.rter.  imr- 
cel  clerk,  stamj)  seller,  etc. 

The  position  of  pc)Stmaster  may  be  njade  an  lionorary  one.  as  a 
reward  for  good  work.  The  postmaster's  duty  may  be  to  clicrk  u|. 
errors  and  sui)ervise  the  office  in  general. 

When  real  letters  are  sent,  allow  the  children  to  get  the  mail 
from  their  boxes. 

Letters  should  be  addressed  only  to  members  of  the  class. 

A  cross  ma<le  with  colored  crayon  may  be  used  for  cancelling. 


S2390 


38 
MINOR  PROJECTS. 

ADDITION  AND  SLBTRACTION.  To  i>lay  trames  learned  iu  a 
lower  yrade. 

To  play  hop-scotch.  A  iliairiain  coutaininff  the  numbers  from  1 
to  lU  is  drawn  on  the  floor.  A  cliild  lioi»8  in  any  direction;  his  score 
lieinir  the  sum  of  the  numbers  hoi»i)ed  ui)on.  If  the  wronjj  sum  is 
•riven,  his  score  is  rejectetl.  Sides  may  be  chosen.  This  may  also  l)e 
played,  scoring  the  difference  of  the  numV)ers. 

To  deliver  letters  to  the  right  house.  One  half  of  the  class  has 
cards  with  numbers  as  5,  7,  S,  etc.,  while  the  other  half  has  cards  with 
combinations  as  9—4,  14—7,  etc.  The  children  having-  answer  cards 
are  at  places  around  the  room.  Others,  the  postmen,  luring  their  let- 
ters to  the  correct  houses.  The  combinations  and  answers  are  read 
as  delivered.     The  children  may  be  score<l  and  timed. 

To  see  who  can  get  rid  of  his  cards  first.  Each  child  is  given 
several  cards,  each  containing  a  number.  The  teacher  calls  out  3+3 
and  all  those  holding  the  card  6  give  it  to  the  teacher.  The  child 
giving  away  all  his  cards  first  is  the  winner. 

To  play  "Tit-tat-toe".  Use  the  well-known  diagram.  The  child- 
ren try  to  find  the  three  numbers  in  a  row,  vertical,  horizontal,  or 
oblique,  which  give  the  largest  sum.  One  child  says  when  it  is  his 
turn.  6,  4,  2,  equals  12.  The  number  12,  with  the  childs  initials,  is 
written  after  the  row  containing  these  numbers.  The  other  child 
says  4,  7,  3,  equals  14.  This  is  recorded,  and  so  on  until  every  one 
is  satisfied  that  the  combination  giving  the  largest  sum  has  been 
found.  X 

X     Adapted  from  The  Decatur  Course  of  Study. 

MULTIPLICATION  AND  DIVISION.  To  capture  the  giant  who 
lives  on  the  hill.  A  hill  with  a  house  at  the  top  is  on  the  board. 
Comliinations  are  on  both  sides  of  the  hill.  Two  children,  one  on 
each  side,  force  their  way  to  the  top  by  calling  the  combinations  cor- 
rectly.    The  child  who  reaches  the  top  first  captures  the  giant. 

To  get  through  the  lines  without  being  stopped.  Children  hav- 
ing c(jmbination  cards  are  guards  on  duty,  their  posts  being  a  good 
distance  ai>art.  Other  children  are  messengers,  attempting  to  get 
through  the  lines,  but  must  first  give  the  i)assword,  i.  e.,  the  correct 
answer  to  the  guard's  combination  card.  If  he  is  not  able  to  give  it, 
he  is  detained  until  he  can. 

To  see  who  can  get  the  most  cards.  The  combinations  are  written 
on  cards,  which  are  shown  by  the  teacher.  The  first  child  telling  the 
sum  correctly  receives  the  card.  The  one  who  receives  the  most 
cards  may  be  the  teacher  for  the  next  game. 

To  keep  out  of  the  center  of  the  circle.  The  teacher  gives  each 
child  a  number  combination.  The  one  who  fails  iu  giving  his  answer 
goes  to  the  center,  but  can  regain  his  place  if  he  succeeds  in  answer- 
ing nK)re  quickly  than   the  one  being  asked. 


39 

To  climb  a  ladiler.  (\)iiil)iiiatit)ii.s  are  written  on  cacli  ruiii,r. 
The  {fame  is  to  see  who  can  climb  hii^hest.  If  a  sec<.ii<l  trial  is  ^ivcn. 
the  child  can  see  how  many  ruu^a  higher  he  can  clinil». 

To  read  the  letters  which  the  postman  brings.  Aisles  are  the 
streets;  seats  are  the  houses.  The  i)ostman  delivers  to  each  house  a 
letter,  combination  card  face  down.  When  all  are  delivered,  the  let- 
ters are  read  as:  2  •  10=-20,  5  -4—20,  8  10^30,  40  •  lo'  4,  etc. 
The  child,  not  able  to  read  his  letter,  fails  to  gfet  another  until  he  can 
and  the  child  who  reads  the  most  letters  in  a  given  time  wins.  De- 
livery is  made  several  times. 

DENOMINATE  NUMBERS. 

These  are  merely  suggestive.  Any  others  which  are  related  to 
the  child's  experiences  and  which  involve  the  required  i)roce8ses  are 
valuable. 

To  plan  for  a  i)icnic  or  j)arty:  the  food  necessary;  the  cost;  the 
amount  an<l  cost  of  the  decorations. 

To  record  the  weight  and  measurement  of  pui)ils,  for  comparison 
later  in  the  year,  noting  the  increases  made. 

To  buy  articles.  Pictures  of  various  objects  with  their  prices  are 
mounted.  Children  make  purchases,  giving  their  sum.  Toy  money 
may  be  used  and  change  given. 

To  show  time  correctly  on  a  i>asteboard  clock  with  nidvable  hands 
such  as,  the  time  for  the  first  bell;  for  Ijeginning  school;  for  reciting 
the  various  suV>jects;  for  recess;  for  dismissal. 

To  <letermine  the  number  of  blocks  walked  in  coming  to  school; 
differences  in  j>upirs  distances;  number  of  blocks  walked  the  entire 
daj';  in  the  five  school  daj's;  by  the  whole  room.  Finding  the  tin)e 
taken  by  various  pupils  in  coming  to  school;  at  what  time  they  should 
start  to  prevent  tardiness,  etc. 

To  determine  the  earnings  and  savings  of  the  whole  class. 

To  make  an  original  i)roblem  from  a  set  of  figures,  as,  8  ■  5,  Jojiti 
has  5  marbles  and  James  has  3  times  as  many.     How  many  has  James? 


Standards  of  Attainment. 

See  (irade  IIT-A. 

Bibliography. 

See  (irade  II I -A. 


40 
(;KA1)K  III-A 


DIRECTIONS. 


U1\1S().\  OK  TI.MH:.  Three-tuurths  (if  the  aritluiieiic  time  allot- 
ment in  thin  irrm\e  should  be  giveu  to  oral  work.  Three-eighths  of 
the  work  should  relate  to  coucrete  problems. 

REXIEW.     The  addition  ami  subtraction  combinations. 
The  tables  of  2,  .5,  10,  3,  and  4. 
IJorrowiufif  in  suV)traction. 
C'arryiuff  in  multiplication, 
C'ouutiny. 

MAIN  TOPICS.  The  fundamental  process.  Emphasis  on  addi- 
tion, subtraction,  and  multiplication. 

See  the  (General  directions  for  (trades  IIB,  IIA,  and  IIIB  relating 
to  the  fundamental  process. 

MULTIPLICATION.  Two-place  nuiltipliers  jireseut  two  difficult- 
ties. 

1 .     Placing  the  second  partial  jiroduct. 
*i.     Adding  the  partial  products. 
E.  g.         2654 

25 

1B270 
5308 


t)6350 
The  first  difficulty  is  most  readily  overcome  by  attempting  little 
or  no  explanation.  The  first  figure  S  is  placed  under  the  multipier  2 
so  therefore  the  S  is  placed  under  the  7.  Place  the  emphasis  on  the 
mechanical  process,  not  on  the  theoretical  explanation.  Require  no 
theoretical  explanation  of  processes  from  the  child.  It  is  enough  to 
expect  him  to  use  them. 


DIVISION.  UsQ  the  long  division  brace  as  5)230  .  This  does 
not  mean,  however,  to  use  the  long  process  for  division  by  one  place 
numl)ers. 

CONCRETE  PROBLEMS.  All  problem  work  in  this  grade  is 
oral.     Only  one-steji  simple  problems  should  be  used. 


SUBJECT  MATTER  AND  PROJECTS. 

NOTATION  AND  NUMER.VriON. 

Counting  by  2'8  to  84. 

Counting  bj-  .S's  to  96. 

Counting  l)y  9'8  to  108. 

Kea<ling  and  writing  numbers  to  100,000. 

The  use  of  the  comma  as  in  14,567. 


41 

ADDITION. 

Rapid  drill. 

Speed  and  time  tests. 

Four  ami  five  jdaee  coluTiiiis  of  six  a<ld('iids. 

^Mixtures  of  four  and  five  place  numbers  as: 
4o(i7  \'1^))\1  84r)(>7 

1.S35  2S17«)  *21.S1> 

1687  19.S84  3r)4()2 

3456  27.S53  :ilS9 

54:}2  24176  4276 

14S()  28.>()4  8.S24(> 

SUBTRACTION. 

Mixed  Problems:  Easy  au<l  Hard  Subtraction.     Trainintr  in  .judt»-- 
ineut  as  whether  to  borrow  or  to  subtract  the  numbers  as  they  staml. 
562  157  42  975 

J.31_  _8S_  1S_  14 

Hard  SuV)tractiou  with  two  stej)  borrowing,  as: 

832  501  30U  9.S21  1870 

148  132  167  1748  483 

Hard  Subtraction,  Mixed,  one  step  and  two  steps,  as: 
721  200  465  619  706 

418  103  158  :US  5S6 

Classification  gfiveu  in  School  Education. 
MULTIPLICATION. 

The  tables  of  6's  and  7"'s. 

Multiplication  by  one  and  two   place  numbers,  the  multiplicand 
to  contain  two  or  three  figures,  and  the  i)roduct  to  contain   not  more 
than  six  digits. 
DIVISION. 

Written  examples  in  short  division  with  borrowing.  I'se  the 
long  division  brace.  The  dividend  should  contain  not  more  than  five 
digits  and  the  divisor  only  one  digit.  Inclu<le  numbers  not  exactly 
divisible. 

FRACTIONS.      ,';  and  \  of    numbers    to    100,    including    numbers    not 
exactly  divisible. 

Adding  and  subtracting  simple  fractions  and  mixed  numbers. 
(No  reductions). 

DENOMINATE  NUMBERS.    One-half  and  one-(juarter  inch. 
The  thermometer.    Small  sub-divisions  need  not  be  used. 

CONCRETE  PROBLEMS.    Addition  of  bills. 
Postage, 
(rarden  jdans. 
Temperature, 
Temj^erature  records. 
OPTIONAL  WORK.    Table  of  8's  and  9's. 
Multiplication  by  three-idace  digits. 
Work  for  excei)tional  sjjeed. 
More  extensive  problem  work. 


42 

GENERAL  PROJECT. 

>This  is  a  sujrtiefsteil  jirojeot  which  will  involve  much  of  the  sub- 
ject matter  of  the  yfiJi'le.  Lest  it  should  uot  provide  sufficient  drill 
to  fix  the  processes,  minor  projects  are  given.  The  latter  are  unre- 
lated, but  miifht  be  an  outgrowth  of  a  larger  project  similar  to  the 
one  here  given. 

rianning  and  making  a  class  garden.  This  project  is  approi*- 
riate  for  the  second  semester  of  the  school  year.  The  planning 
ef  the  garden  maj-  be  begun  as  early  as  February.  The  early  weeks 
of  the  second  term  may  also  be  utilized  for  the  planting  of  window 
boxes  so  that,  as  soon   as  the  weather  permits,  plants  may  be  set  out. 

In  the  suburbs,  it  will  not  be  difficult  to  secure  a  lot  large  enough 
for  forty  miniature  ganlens.  In  a  thickly  populated  district,  more 
<lifficulty  may  be  experience<L  Very  often,  however,  a  patch  of  school 
property  adjt>ining  the  building  may  V)e  available  for  a  garden. 

There  are  certain  exjieuses  connected  with  a  problem  of  this 
nature.  In  some  cases  the  land  must  be  fertilized  and  plowed.  Then, 
too,  seeds  must  be  purchased.  It  may  even  be  necessary  to  ijurchase 
a  few  garden  implements,  although  most  of  the  children  will  be  able 
to  bring  these  from  home.     How  shall  these  expenses  be  paid  ? 

hleveral  suggestions  present  themselves,  (l)  The  class  may  earn 
the  money  by  collecting  and  selling  rags,  newspapers,  magazines  and 
tin  foil.  (2)  In  some  schools  it  may  be  possible  to  hold  a  candy  sale. 
(3)  The  class  may  give  an  entertainment,  charging  a  small  admission 
fee.  (4)  Pennies  earned  by  running  errands  or  selling  papers  may 
be  added  to  the  garden  fund. 

This  garden  project  facilitates  a  correlation  of  Arithmetic  with 
Nature  Study  and  Industrial  Arts. 

The  following  are  merely  suggestions  showing  how  a  practical 
application  of  1 1 1- A  Arithmetic  processes  may  be  made  in  planning 
the  garden: 

If  each  of  the  forty  members  of  the  class  should  earn  25  cents 
for  the  garden  fund,  how  much  money  would  we  have  V 

Tin  foil  sells  for  40c  a  pound.  We  have  eleven  pounds,  IIow 
much  will  that  sell  for?  If  we  need  85.00  to  pay  for  fertilizing  and 
plowing  the  garden,  how  much  more  money  must  we  earn  ? 

Newspai)t-rs  sell  at  the  rate  of  100  pounds  for  25c.  How  many 
pounds  would  we  have  to  sell  to  secure  81.00  ? 

Small  garden   hoes  cost  35c  a  piece.     IIow  much  will  four  cost  ? 

The  garden  lot  affords  splendid  opportunity  for  making  measure- 
ments.    The  following  problems  in  mensuration  suggest  themselves: 

Find  the  length  of  the  lot  in  yards.      IIow  many  feet  is  that? 

L'^sing  a  ruler,  now  measure  the  length  of  the  ground  in  feet. 
How  does  this  numl)er  comi)are  with  your  first? 

Find  the  width  of  the  lan<l  in  yards  and  feet.  How  much  greater 
is  the  length  tiian   the  width  ? 

What  is  the  distance  around  the  entire  lot  in  feet?  What  would 
be  the  distance  around  one-sixth  of  this  lot?     One-seventh? 

Assuming  that  the  lot  may  be  50  feet  wide  and  <S0  feet  long,  each 
member  of  the  class  of  40  might  then  be  given  a   small   plot   ten   feet 


43 

S(iuare.  Childreu  in  the  A  Tliird  (irade  could  not  work  this  out  for 
themselves  as  scjuare  root  is  involved.  However,  after  the  class  has 
been  told  that  each  meinher  may  have  an  individual  {rarden  ten  feet 
on  a  sitle,  they  can  easily  measure  off  the  ground  themselves. 

Letting  one-inch  stand  for  two  feet,  draw  a  plan  of  your  jjanlen. 
What  measure  on  your  ruler  will  you  use  to  show  a  foot?  a  half-foot? 
Paths  should  be  one  foot  wide.  In  makiny  your  yfarden  jdan,  show 
what  you  intend  to  plant  in  each  bed. 

In  regard  to  seeds,  the  childreji  may  j^rice  these  at  different 
places,  viz.:  the  grocery  store,  florist  shoj)  and  seed  store.  They  may 
then  compare  costs.  If  all  seeds  are  purchased  at  one  time  a  re- 
duction may  be  obtained.  The  purchasing  of  seeds  may  involve  such 
problems  as  these: 

If  Mr.  A.  sells  lettuce  seeds  at  <S  cents  a  i)ackage,  and  Mr.  ]>.  the 
same  at  (J  cents  a  package,  what  will  be  the  difference  in  price  if 
seven  packages  are  purchased? 

If  one  package  of  seeds  will  produce  fifty  head  of  lettuce,  how 
many  heads  will  twelve  packages  produce?  What  will  twelve  jiack- 
ages  cost  at  six  cents  a  package? 

Beans,  and  peas  are  sold  at  loc  a  half  pint.  Find  the  cost  of  a 
quart. 

§!4.50  have  been  earned  for  seeds.  How  many  packages  will  this 
buy  at  eight  cents  a  package? 

Account  must  be  kept  of  all  seeds  pur'fhased.  How  much  more 
money  was  spent  for  peas  than  for  radishes?  How  much  less  was 
spent  for  carrots  than  for  beans? 

Temperature  problems  may  also  form  part  of  this  project,  e.  g., 

What  is  the  freezing  point  on  the  themometer?  How  shall  we 
protect  our  plants  when  thermometer  reaches  82  degrees  above  zero? 

At  what  time  of  day  is  the  mercury  low?  When  is  it  highest? 
Why  should  we  not  work  in  the  garden  at  noontime? 

How  much  below  the  freezing  point  is  12  degrees?  What  is  the 
difference  between  6S  degrees  and  98  degrees? 

The  children  should  keep  account  of  all  time  spent  working  in 
their  gardens. 

If  it  takes  a  child  90  minutes  to  hoe  6  rows  of  beans,  how  long 
will  it  take  him  to  hoe  one  row? 

If  one  child  destroys  47  weeds  in  a  niorniiig.  how  many  wee<ls 
will  40  children  destroy? 

If  each  child  spends  20  minutes  a  day  in  the  garden,  how  much 
time  will  40  children  sjtend?  How  many  minutes  will  this  amount  to 
in  a  week?     In  a  month? 

Close  account  should  also  be  kejjt  of  all  i)rodiu-ts  sold,  'i'liis 
phase  of  the  projects  points  toward  such  jn-oblems  as  these: 

If  cabbage  sells  for  fifteen  cents  a  head,  how  much  will  be  .leriv- 
ed  from  the  sale  of  86  heads? 

Tomatoes  sell  for  Kic  a  i)ound.  How  much  is  that  an  ounce? 
How  much  would  seven  ounces  sell  for?     Seven  pounds? 

r)4c  was  spent  for  lettuce  seeds.  The  lettuce  sold  for  n2.).  1- ind 
the  profit. 


44 

If  the  pi-Ktit  ou  peas  anmuiitt*  to  *l«).4:i  wliile  that  ou  carrots  is 
*19.5t).  tin«l  the  differeuce. 

If  peas  sell  for  6f  a  (luart;  liow  many  (luarts  must  be  sohl  for  !i!l.50. 

Frai-tioiis.  too,  may  euter  the  class  j^ardeii. 

(.'utworms  destroyeil  7  of  the  tomato  i)laiits  in  the  garden.  If 
there  were  4'i  jilaiits  set  out.  how  many  were  destroyed. 

If  jieas  sell  for  x*2.00  a  bushel,  how  much  should  be  chartjed  for 
\  bushel? 

REMARKS: 

This  i»roject  may  be  introduced  into  the  Nature  Study,  Industrial 
Art.s  ami  I^aniruaire  lessons  as  well  as  in  the  Arithmetic  work. 

In  the  Nature  study  class  the  children  may  exi»eriment  with  diff- 
erent kinds  of  soil  to  find  out  which  is  the  most  suitable  for  cultiva- 
tion of  certain  vegetables.  In  this  class,  too,  the  time  of  plantiufj: 
should  be  discussed,  for  the  i^rospective  yardeners  must  know  which 
seeds  to  plant  first.  Account  should  be  kept  of  the  number  of  days 
that  pass  before  each  vegetable  is  ready  for  gathering. 

In  the  Language  class  the  children  may  write  compositions  about 
different  phases  of  gardening.  A  study  of  such  masterpieces  as  The 
Sower  or  Plowing  will  be  enjoyed  at  this  time. 

This  project  may  also  be  used  to  motivate  Industrial  Arts  lessons. 
Appropriate  garden  booklets  may  be  made  containing  crayon  draw- 
ings, water  color  sketches,  and  paper  cuttings.  C'omjjositions  written 
in  the  Language  class  may  have  a  place  in  this  booklet.  Attractive 
garden  posters  may  also  be  made. 

Each  chihl  shouhl  keep  strict  account  of  money  expended  for 
seeds,  time  spent  in  the  garden  daily,  and  amount  realized  from  sale 
of  products.  Any  products  that  are  used  in  the  home  should  be  list- 
ed as  sold  in  the  account  book.  These  rei)orts  may  be  used  in  the 
Sejitember  work. 

During  the  summer  months,  garden  supervisors  are  appointed  by 
the  IJoard  of  Education  to  visit  school  gardens  and  give  the  children 
advice  as  to  the  care  of  their  gardens  and  the  disposal  of  their 
products. 

MINOR  PROJECTS. 

ADDITION  AND  SUBTRACTION.  To  play  any  of  the  games 
learned  in  a  lower  grade. 

To  see  which  row  can  score  highest.  The  numbers  from  1-20 
are  written  on  the  board.  One  score-keeper  from  each  row  is  ap- 
pointed. The  leader  iM)int8  to  a  number  on  the  board.  The  first 
cliild  in  the  first  row  gives  the  number  which,  added  to  the  number 
pointe<l  to,  will  make  20.  Each  child  may  have  8  or  4  chances.  The 
game  is  continued  until  all  have  had  a  chance.  The  row  having  the 
liighest  total  score  wins. 

To  ad<l  iiuml>ers  correctly,  as  fast  as  the  teacher  gives  them. 
Children  are  at  their  seats;  the  teacher  tells  them  to  begin  with  a 
certain  nunil»er  as  "2,"  then  the  teacher  says  "add  3,"  the  chihlren 
writing  .o,  then  "add  (3,"  the  children  writing  11, 


45 

To  i)lay  JJasket  IJall".  Sides  are  t-liosen.  A  line  upon  whicli  llic 
players  stand  is  drawn  seven  feet  from  the  \vaste-l)asket.  TLey  throw 
beau  batrs  into  the  basket,  each  bag  counting-  a  nuinl)er  of  j)oints. 
Several  turns  may  be  given.  The  score  is  the  sum  of  all  bags  thrown. 
Two  score-keepers  score  for  their  sides.  Adapted  from  the  Decatur 
Course  of  Study. 

To  keep  out  of  the  tisli  pond.  The  class  forms  a  cii'clc.  Kach 
child  is  given  a  combination.  If  he  cannot  give  the  answer  correctly, 
he  becomes  a  fish  and  must  stand  in  the  center.  He  can  get  out  by 
giving  some  one  else's  combination  l)efore  they  do,  thus  exchanging 
places. 

MULTIPLICATION  AND  DIVISION.  To  avoid  l)eing  caught  by 
the  fox.  Children  form  a  big  circle.  The  fox  calls  a  combination 
asking  a  goose  to  give  the  answer.  I'nless  he  answers  correctly,  he 
is  caught  and  becomes  a  fox.  The  geese  may  regain  their  places  by 
giving  the  combiuatious  more  (juickly  than  the  geese  in  the  circle. 
Foxes  permitting  a  wrong  answer  to  stand  as  correct,  may  be  caught 
by  the  geese. 

To  play  Snow-man.  A  snow-man  is  drawn  upon  the  board  and 
snow-balls  each  containing  a  combination  are  placed  beside  him.  ^V 
child  pretends  to  pick  up  a  snow-ball,  at  the  same  time  giving  an 
answer.     If  the  answer  is  correct,  part  of  the  snow-man  is  erase<l. 

To  play  the  game  of  I>ean  Bags.  Three  circles  of  different  sizes, 
each  having  a  different  value  as  7,  <S,  9  are  drawn  on  the  floor.  Two 
leaders  choose  people  for  their  sides.  Each  child  throws  two  bags, 
the  score  being  their  sum  or  product.  The  score  for  each  side  is  de- 
termined Avhen  all  have  had  a  turn.  Adapted  from  the  Baltimore 
Course  of  Study. 

To  see  who  can  get  rid  of  his  cards  first.  Cards  containing  the 
combinations  are  <listributed  evenly  to  the  class  which  sits  in  a  circle. 
The  first  child  holds  up  a  card,  calling  on  another  child  to  give  the 
answer.  If  that  chihl  does  not  know,  he  must  take  the  card,  but  if 
he  does,  the  first  child  keeps  the  card.  The  next  chihl  then  has  a 
turn,  and  so  on  until  one  child  is  out  of  cards.  That  person  is  winner. 

To  gain  a  greater  number  of  cards  than  an  opponent.  Comldna- 
tion  cards  are  placed  along  the  blackboard.  Two  children  starting 
at  opposite  ends,  capture  combination  cards  by  railing  them  correctly. 
The  child,  having  the  most  at  the  end  of  one  minute,  wins. 

DENOMINATE  NUMBERS. 

Suggestive  Only 

To  keep  accounts  including   amounts  ami    costs   for    a    dctinite 
length  of  time  of  the  following: 
Selling  jiajiers. 
Delivering  milk. 
Collecting  scrap. 
Kuniiing  errands. 
Ibiying  '^rhrift  Stamjts. 
Recording  the  cost    of    pencils,    jtajiers,    iiem-il    boxes,     erasers, 
etc.,  bought  for  school  use. 


46 

Keei»intr  a  record  of  the  outside  temperatures  for  a 
week  or  month. 

Estimating  the  amount  ami  cost  of  seeds  needed  in  mak- 
intr  a  trarden.  Checking  the  selling-price  of  the  vegetables 
in  the  fall. 

Finding  the  postage  necessary  for  a  number  of  letters. 
Five  letters  and  three  postcards. 

Standards  of  Attainment. 


Woody  test;  Series  A. 

Addititm 

14.5 

Subtraction 

11.2 

.Multii»icatiou 

4.7 

Division 

5.8 

Courtis  Test;  Series  B,  Speed. 
A<l<lition  4 

Subtraction  5 

Courtis  Test;  Series  B,  Accuracy. 
Addition  41 

Subtraction  49 


Bibliography. 

TEACHER'S  READING: 

Brown  and  Coffman:  IIow  to  Teach  Arithmetic. 

Chap.  VIII  The  Value  of  Drill. 

Chap.  X  Primary  Arithmetic. 

Cliap.  XI  Teaching  the  Fundamentals. 
Kendall  and  Mirick:  IIow  to  Teach  the  Fundamental  Subjects. 

Pp.  10.S-lS(j  Mathematical  Skill. 

Pj..  195-200  Drill-Tests. 
Klapper:  The  Teaching  of  Arithmetic. 

Chajt.  Ill  and  IV  General  Principles. 

Chap.  VII   The  Fundamental  Oi)erations. 
Sn)ith:  The  Teaching  of  Arithmetic. 

Chaii.  Xn'   Details  for  Exi)eriment. 

Chaj).  XVII    Work  of  the  Tliird  School  Year. 
Strayer  and  Xorsworthy:   How  to  Teach. 

"pp.  204-205  The  Drill  Lesson. 
Suzzalo:  Tlie  "Teaching  of  Primary  Arithmetic. 

Chap.  IX  Sjtecial  Methods  for  Obtaining  Accuracy  and  Speed, 
Wilson  and  Wilson:  Motivation  of  School  Work. 

Chap.  IX   Motivation  of  Aritlnnetic. 

SUPPI.I:M KNTARY  BOOKS: 

II<>\t  and  Peet:   Everyday  Arithmetic,  Book  I. 

Stunt-  Millis:   Primary  Arithmetic. 

Thoriidike:   .\rithiiietic.  Book  One. 

Weill wortli  and  Smith:   Essentials  of  Arithmetic,  Primary  Book, 


DIRECTIONS. 


47 
GKADK   IV-i;. 


DIVISION  OF  TIME.  Thrt'e-f..urtlj.s  ,.f  all  the  aiitliiiR-tir  time 
allotnieut  sliDuld  l)e  given  to  oral  work.  Most  of  tlie  work  should  he 
abstract. 

REVIEW.   Multiplieatiou  by  8,  4,  5,  G  an.l  7. 
Part  takiug-. 

MAIN  TOPICS.  The  four  fundamental  i)rocesses.  Emi)hasis  on 
niultii>lieati()n. 

This  grade  is  especially  suited  to  mechanical  drill.  Abstract  work 
is  performed  with  interest,  without  regard  to  its  concrete  api»licatioii. 
The  desired  aim  is  speed  and  accuracy  in  all  the  fundamental  opera- 
tions. The  use  of  standard  tests  provides  an  incentive  to  each  indi- 
vidual to  increase  efficiency  in  the  fundamentals.  Only  such  work 
as  cannot  be  done  mentally  should  be  recorded  on  paper.  Various 
types  of  oral  work  may  be  given  as  follows: 

Projects  involving  work   on   the   combinations   in  aildition, 

subtraction,  multiplication  and  division. 

Projects  involving  measures  and  their  ajjplication. 

Problems  without  figures. 

Problems  with  small   numbers,  emphasizing   one   process  in 

the  solving  of  one-step  concrete   problems.     The  amount  of 

work  done  on  concrete  problems  need  not  be  large. 
Make  the  children  independent  by  teaching   them   to  check  and 
verify  their  ow^n  results  as  follows: 

Check  addition  by  aibling  the  columns  in  reverse  order. 

Check  subtraction  by    adding   subtrahend  and  difference  to 

give  the  minuend. 

Check  multiplication  by  multiplying  a  second  time  until  di- 
vision is  understood. 

Check  division  by  multiplication. 
The  review  work  need  not  be  disheartening  for  the  process  of 
re-learning  is  much  easier  than  the  original  process.  Avoitl  going 
into  the  advanced  work  before  the  previous  work  is  sufficiently  at 
command.  If  it  is  assumed  that  the  pupils  have  forgotten  much  of 
the  previous  work,  discouragement  on  both  the  teacher's  and  pupil's 
]»art  will  be  avoided. 

ADDITION.  Permit  no  eouutiiig  or  hesitancy  in  giving  combin- 
ations. If  the  child  hesitates,  tell  him  the  answer.  Center  attentit)n 
on  the  combinations  which  present  difficulty  to  your  class.  The  fol- 
lowing columns  contain  all  the  combinations.  They  may  be  placed 
on  the  board  ])ermanently  and  used  for  time  tests. 


0 

o 

1 

4 

!) 

!» 

s 

2 

2 

5 

7 

9 

!» 

7 

•  > 

•6 

8 

4 

r. 

1 

■") 

() 

1 

4 

4 

S 

() 

1) 

8 

•I 

."> 

8 

8 

o 

2 

( 

7 

s 

1 

2 

<) 

4 

5 

(> 

1 

s 

H 

48 

MULTIPLICATION.  Clear  n]>  dittifulties  an<l  strentftheu  individ- 
ual \veaknes.>*ej?  iu  multiiilic-atioii  !>>  two  tiuuifs  before  iiiultii)licatiou 
l»y  tliree  ti«:ures  is  atteinjiteil. 

DIVISION.  I'se  Itiiiir  ilivi.xiou  brace  for  the  short  division 
process.  Ill 

(j)  <)<)() 

FRACTIONS.  The  subject  of  fractions  as  a  definite  topic  is 
taken  u\>  in  tlie  Fifth  (iraile.  In  the  Fourtli  (rrade,  keej)  the  work 
simple  and  as  much  as  i)ossible.  ol)jective. 

CONCRETE  PROBLEMS.  Aim  for  thorough  mastery  of  the  one- 
step  problems  in  this  yfrade  so  that  the  child  will  know  exactly  when 
to  atld,  subtract,  multii»ly  or  divide.  For  this  much  practice  must  be 
triven.  If  lartje  numbers  seem  to  confuse  the  child,  f>ive  the  same 
tyi»e  of  ]>roblems  usiny  smaller  numbers. 

DECIMALS.  ]i,  reading:  U.  S.  money  read  the  decimal  point  as 
and.     -s24..')0  is  to  be  read.  "Twenty-four  dollars  and  fifty  cents. 

SUBJECT  MATTER. 

NOTATION  AND  NUMERATION. 

Iieadin<>-  and  writing-  numbers  to  1,000,000. 

Counting'  to  100  V>y  ll\s  and  12's. 
ADDITION. 

Furthei-  drill  on  addition. 

Increased  length  of  columns-seven  addends. 
SUBTRACTION. 

Hard  subtraction,  three  steps,  as  — 

«001  742S  40009         •  400i»l 

5783  6679  12078  14829 

liapid  drill. 

Si)eed  ami  time  tests. 
MULTIPLICATION. 

Tables  of  8''s  and  9's. 

.Multii»liers  of  two  and  three  digets. 

Zero  in  the  multiplier. 
DIVISION. 

Reverse  of  multiplcation. 

Rapid  <lrill. 

Si)ee<l  and  time  tests. 
DENOMINATE  NUMBERS. 

'I'on.  peck  and  bushel 
FRACTIONS. 

I'art  taking-. 

K  and  9  of  any  two  place  numbers. 

Addition    and    subtraction    of    simi>le    fractions    and   mixed 

numbers.      NO  leductions. 
DECIMALS. 

r.  S.  money  correctly  uiitten  and  read. 

Correct   use  of  "'ainl.'" 


49 
GENERAL  PROJECT. 

This  is  a  su^yested  jji-oject  whicli  will  involve  niucli  of  tlie  suli- 
ject  matter  of  the  yrade.  Lest  it  should  not  provide  sutticieiit  drill 
to  fix  the  processes,  minor  projects  arc  yiveu  as  details  under  each 
topic  ot"  subject  matter.  The  latter  are  unrelated  but  niitjht  be  an 
outgrowth  of  a  largfer  project  similar  to  the  one  here  given. 

Uuying  Coal.  This  project  necessarily  implies  a  correlation  (»f 
Arithmetic  and  (geography.  In  the  Geography  class  the  children 
may  trace  the  route  followed  by  vessels  bringing  coal  from  Erie  to 
Duluth.  ()l)servatiou  trips  to  the  <locks  during  the  uidoadiiig  of  coal 
barges  will  vitalize  the  geography  lessons. 

In  developing  this  project,  the  following  idoblems  eiiilMnlying 
Arithmetic  processes  to  be  emphasized   in  IV-IJ  are  suggested: 

1.  N'essels  bringing  coal  from  Erie  to  Duluth  consume  30  tons  of 
vval  a  day.  How  many  pounds  is  this?  It  takes  ten  days  to  make 
the  rouml  trip.  How  many  tons  are  consumed  during  the  trip?  At 
§3.00  a  ton  what  would  this  amount  to? 

2.  A  freight  boat  travels  at  the  rate  of  nine  miles  an  Ikmii-.  jjuw 
far  will  it  travel  in  twenty-four  hours  ?  in  five  days. 

3.  I)oats  which  bring  us  our  coal  travel  1,02.5  miles.  Find  one- 
eighth  of  this  distance;  one-ninth. 

4.  It  takes  one  hour  to  unload  1.000  tons  of  coal.  How  many 
tons  can  be  unloaded  in  ten  hours? 

5.  A  vessel  brings  7,000  tons  of  coal.  If  the  coal  sells  at  -sit. (Hi  :i 
ton,  how  much  is  the  cargo  worth? 

The  children  may  bring  to  school  samjdes  of  various  kinds  of 
coal.  To  each  sainple  a  tag  might  be  attached  giving  the  price  per 
ton.     This  will  suggest  problems  of  the  following  nature: 

Which  kind  of  coal  is  most  expensive? 

How  much  more  does  a  ton  of  nut  coal  cost  than  pea?  What 
would  be  the  difference  in  price  in  buying  ten  tons  of  either  ? 

If  a  ton  of  stove  coal   costs  nine  dollars,  find  the  cost  of  4S  tons. 

Find  the  total  cost  of  a  ton  of  bii(iuets,  a  ton  of  screenings,  a  ton 
of  nut  coal  and  a  ton  of  i>ea. 

The  pupils  may  find  out  from  their  jtarents  the  cost  of  the  tuel 
burned  in  their  homes  during  the  winter.  Encourage  tlic  childien  to 
think  of  ways  of  saving  fuel.  e.g.  (1)  by  using  storm  windows,  (2) 
by  closing  all  cracks  and  crevices,  (3)  by  keeping  outer  doors  locked. 
(4)  by  cleaning  out  the  furnace  or  stove  at  frequent  intervals.  (•'))  by 
shutting  off  the  heat  in  rooms  which  are  not  in  use. 

Comparisons  may  be  made  as--- 

At  Mary's  house'the  fuel  bill  was  .sOO.()S;  at  .l..lni">  ii  was  >;lii().4:{. 
Fin<l  the  difference. 

Mr.  A  bought  19  tons  of  coal  an. I  Mr.  T.  14.  Find  tlic  dift.Tcnc*' 
in  tons;  in  i)ounds. 

In  an  eight-room  house  19.14;')  jiounds  of  coal  wert-  (•oiisuim-d. 
How  many  i>ounds  were  required  to  heat  one  room  ? 

From  an  observation  lesson  the  childien  ina\-  h-arn  li<>w  c<>al  is 
weighed. 


50 

A  loa«le«l  coal  watrou  weighs  10,900  pouudf*.  If  the  team  alone 
weiifhs  5,900  iioumls,  how  much  does  the  coal  weigh  'i 

A  coal  team  weighs  (>,'200  pounds.  Two  tons  of  coal  are  loaded 
onto  it.     Fiinl  the  weight  of  the  team  and  coal  together  in  pounds. 

A  man  can  loa<l  ti.OOO  pounds  of  coal  in  an  hour.  How  many 
piMunIs  can  he  load   in  S  hours?  in  9  hours? 

I4S.7tjO  ]iouuils  of  coal  are  to  be  placed  on  8  equal  loads.  How 
many  iiouuds  in  each  load? 

Following  are  problems  V)ased  on  tlie  chief  engineer's  rei)ort  of 
coal  consumed  in  Duluth  schools  during  1917-18: 

The  fuel  l»ill  for  the  Bryant  School  was  §1,549.51;  for  the  Fair- 
mount  *908.88;  for  the  Franklin  xl, 147.49;  for  the  Lowell  8650.39. 
Find  the  total  amount  for  these  four  schools. 

1GO,U50  jtounds  of  screenings  and  '286,500  pounds  of  soft  coal 
<lust  were  purchase<l  for  the  Jackson  school.  How  many  more  pounds 
of  dust  were  useil  than  screenings? 

The  fuel  bill  for  the  Madison  School  amounted  to  8788.77.  This 
was  8'i9.S.18  less  than  the  bill  for  the  Longfellow.  What  was  the  bill 
for  the  Longfellow  School  ? 

Fuel  purchased  for  the  Monroe  School  amounted  to  8712.80,  for 
the  Salter  8881.76,  for  the  Webster  8408.78,  for  the  Lester  Park 
^875.05  and  for  the  Central  High,  810,551.07.  How  much  more  was 
spent  for  the  Central  High  School  than  for  the  other  four  together? 
How  much  less  was  expended  for  the  Monroe  School  than  for  the 
Lester  Park  ? 

REMARKS:  This  project  may  be  correlated  with  the  Language 
lessons.  After  a  visit  to  the  coal  docks,  the  class  may  write  original 
compositions  on  the  unloading  of  coal,  number  and  position  of  docks 
and  flistribution  of  the  coal  to  points  in  the  Northwest.  These  com- 
positions may  be  put  in  booklets  made  during  the  Industrial  Arts 
jteriod.  Such  booklets  may  contain  appropriate  drawings  and  paper 
cuttings,  e.g.  a  coal  barge  entering  the  canal. 

MINOR  PROJECTS. 

ADDITION  AND  SUBTRACTION.  To  solve  live  problems  cor- 
rectly within  a  given  amount  of  time. 

Problems  in  aiMition  and  subtraction  are  given,  the  child  start- 
ing and  stopi)iiig  on  signal.  Individual  scores  are  kept  on  the  boanl 
from  ilay  to  day.  the  child  noting  his  imi)rovement. 

To  add  and  subtract  numbers  as  fast  as  the  teacher  can  give  them. 
The  riietho<l  is  a  follows: 

Tkachkk:  liegin  with  8,  add  6;  (children  write  14);  sub- 
tract 5;  (children  write  9)  etc.,  using  the  difficult 
combinations. 

To  run  a  race.T  he  combinations  in  addition,  subtraction,  multi- 
plication, or  division  are  printed  on  cards,  sufficiently  large  so  that  all 
the  class  can  see.  A  correct  answer  counts  1;  an  incorrect  answer  0. 
A  score-keeper  is  aj)pointed.  The  game  is  to  see  which  row  scores 
highest. 


51 

To  write  the  correct  iiumlters.  A  cliilil  «joes  to  tlie  teadier's  ilesk 
selectiujj  three  tickets,  each  coutaiuiuu"  a  number  from  0-10.  Wliile 
he  adds  the  class  listens  aud  writes  the  numbers  which  must  have  been 
taken.  For  instance,  if  the  child  chooses  5,  6,  7,  he  says  5,  11,  IS.  and 
the  class  writes  5,  6,  7.  Those  at  their  seats  cheek  to  see  how  many 
combinations  they  had  were  correct.  After  a  little  jtractice,  more 
than  three  numbers  may  be  chosen. 

MULTIPLICATION  AND  DIVISION. 

To  guess  numbers  which  make  a  certain  product. 

"I  am  thinkinfj'  of  a  number  between  30  and  40." 

"Is  it  6     6?" 

''No,  it  is  not  36." 

"Is  it  7-5?" 

"Yes,  it  is  35." 

Adapted  from  the  Decatur  Course  of  Study. 

To  race.  A  list  of  combinations  is  written  upon  the  V)oard.  One 
child  starts  aud  names  the  results  quickly.  If  an  error  is  made,  the 
child's  name  is  written  at  that  place  and  another  child  begins  at  the 
beginning.  He  goes  as  far  as  he  can  and  his  name  is  written  where 
he  stops.  If  a  child  finishes,  his  name  is  written  at  the  side  as  a 
winner.  A  child  who  missed  may  have  another  chance,  beginning  at 
the  place  where  he  made  a  mistake. 

To  see  which  leader  can  win.  Three  leaders  are  chosen  each 
being  given  five  cards  containing  combinations.  Each  in  turn  hoMs 
up  a  card  aud  calls  on  some  one  for  a  (juick  answer.  The  child  an- 
swering correctly  joins  that  leaders'  group.  The  leader  who  gels  the 
largest  number  iu  his  group  wius. 

To  race.  The  teacher  writee  an  uneven  number  of  examples 
across  the  board.  One  child  begins  to  work  at  the  right,  the  other  at 
the  left.  The  one  who  reaches  the  middle  example  first  has  the  priv- 
ilege of  working  it  and  wins  the  race. 

To  race.  The  pui)ils  draw  circles  on  the  Ijlackboard.  entering 
the  numbers  from  1  to  9  inside  the  circumference.  Each  child  is 
given  a  number  to  i)lace  iu  the  center  of  his  circle.  The  game  is  to 
see  who  can  first  write  the  products  of  the  inner  number  and  the  out- 
er numl)ers  on  the  outside  of  the  circle. 

To  i)lay  Bean  Bag.  Two  concentric  circles  are  drawn  on  the 
floor.  The  outer  circle  is  divided  into  quarters.  Write  a  figure  in 
the  inner  circle  and  in  each  (juarter  of  the  outer  circle.  The  child 
scores  the  sum  of  the  number  hit  in  the  outer  circle  times  the  inner 
number.     The  scoring  is  done  by  groups. 

Adaptetl  from  the  Decatur  Course  of  Stuily. 

To  play  Heturn  Ball.  Draw  a  large  oblong  on  the  blackboard. 
Mark  it  off  into  a  number  of  squares  and  triangles,  writing  a  number 
iu  each  si)ace.  The  children  use  a  return  ball  and  try  to  strike  the 
largest  number  in  a  si)ace.  If  the  jdayer  strikes  a  number,  lie  is  al- 
lowed to  multiply  b\-  o.  or  any  number. 

Adapte<l  from  the  Decatur  Course  of  Study. 


52 

DENOMINATE  NUMBERS.  FRACTIONS  AND  DECIMALS.  To 
fiii.l  the  relatiou  ut"  the  pec-k  t..  the  bushel  tlirouirli  experimeutation 
with  linth. 

To  riii<l  out  the  i.ro.lufts  wliich  are  sold  by  the  peck  or  bushel. 

To  obtain  the  jiriees  for  such  i»ro<luee. 

To  make  oriyiual  jtrobleius  relatiujj  to  the  above,  using  %ures 
whii-h  are  accurate. 

To  fiud  the  cost  iier  ton  of  various  kiuds  of  coal. 

To  tiii.l  the  amount  of  coal  needed  for  family  use  during  the 
winter. 

To  find  the  cost  of  all  the  coal  needed  during  the  year. 

To  find  the  cost  of  clothing  needed  by  boy  or  girl  during  the  year. 

T<t  solve  jiroblems  relating  to  the  rainfall  and  temperature  of 
Duluth: 

Kainfall  Temperature 

(by  inches)  C^Fahreuheit) 

Dec.  1  IT 

Jan.  1  10 

Feb.  1  13 

Mar.  1',  24 

Apr.  2  38 

May  3  48 

June  4  58 

July  4  65 

Aug.  3  64 

Sept.  3.^  56 

Oct.  3i  45 

Nov.  V,  29 

Finding  the  total  rainfall  for  Duhith;  amount  of  rainfall  in  win- 
ter, spring,  summer,  fall;  diftereuce  in  rainfall  V)etween  any  two 
months. 

Finding  the  hottest  month:  the  coldest:  the  difference  between 
the  two;  difference  between  any  two  months. 

To  solve  problems  relating  to  industries  of  Duluth. 

There  are  four  ore-docks.     Each  has  384  ore  pockets. 

I  low  many  pockets  are  there  in  allV  The  ore  is  brought 
down  from  the  mines  in  trains  of  50  cars,  each  car  holding 
fifty  tons.  How  many  tons  of  ore  are  brought  down  in  a 
trip?  etc.  For  more  statistics,  see  the  pamphlet  on  Duluth 
issued  by  tlie  Commercial  Club.  \t.  46. 


58 

Other  subjects  which   are  suffjfestive: 

P(>l»ulati<)U  in  various  years. 

Public  Schools. 

Number  of  children  in  school. 

Miles  of  i»aveil  street. 

Parks. 

Iron  Ore. 

Coal 

Number  of  churches. 

Books  in  the  library,  etc. 

To  solve  problems  relating  to  travel  by  airjtlaue. 

Distance  across  ocean. 

Time  taken  to  ascend;  to  descend. 

Temperature  at  various  levels. 

Speed  of  aeroplanes,  etc. 

Time  taken    by  lieid    to  cross    in  a    hydroplane;  by    IJrown 

and  others  in  a  non-stop  flight. 

To  solve  problems  relating-  to  travel  by  water. 

Length  of  boats. 

Speed. 

Number  of  passengers. 

Cost  per  trij)  and  jter  mile. 

Numlier  of  deck-hands,  etc. 

Standards  of  Attainment. 

See  (irade  IV- A. 

Biblio^aphy. 

See  Grade  IV- A 


54 

(;kade  iv-a. 


DIRECTIONS. 


DIVISION  OF  TIME.  Three-fourths  of  the  arithmetic  time  allot- 
ment ill  this  irrade  should  l»e  giveu  to  oral  work.  Most  of  the  work 
should  l»e  with  aV)Stract   iiumhers. 

REVIEW.    Part  takiuj?.     Multiplicatiou  l>y  (j,  7.  S  aud  9. 

LEADING  TOPIC.      Loutr  Divisiou. 

IJead  the  <lireetioiis  giveu  for  preeediuy  grades,  especially  (xrade 
4-B.  The  poiut  of  emphasis  iu  this  half  jjrade  is  long- divisiou.  "This 
necessitates  daily  drill  on  the  multiplication  tables  and  upon  such 
questions  as,  How  many  nines  iu  67  ?  in  75  ?  in  71? — as  a  preparation 
for  division.  Long- division  should  be  approached  through  analyzing 
what  has  been  done  by  pupils  in  short  division,  and  finding  that  four 
steps,  repeate<l  over  an<l  over  again,  are  all  that  must  be  known. 
These  steps  are: 

ESTIMATE  how  many  times  the  divisor  can  be  fouml  in  the  par- 
tial dividend  necessary  to  use. 

MULTIPLY  the  divisor  by  the  estimate  made. 

SUBTRACT  the  pro<luct  of  the  divisor  and  quotient  figure,  or  es- 
timate, from  the  i)artial  dividend  used. 

BRING  DOWN  the  next  figure  of  the  dividend,  to  be  placed  at  the 
right  of  the  remainder,  all  of  which  now  becomes  a  new  partial  divi- 
dend with  which  the  steps,  estimate,  multiply,  subtract,  and  bring- 
down,  are  again  used. 

When  the  steps  are  once  secured  from  an  analysis  of  a  short  di- 
vision example,  pupils  should  test  their  use  of  the  steps  Avith  divisors 
like  21,  71,  31,  51,  61,  so  that  the  ones'  digit  is  sure  to  change  the 
quotient  estimate,  even  to  the  extent  of  needing  to  think  40  for  39, 
30  for  i!9,  70  for  69,  etc.,  in  making  the  estimate.  After  this  the  di- 
visors should  be  32,  52,  92,  and  so  on;  78,  68,  28,  etc.;  97,  47,  etc.;  un- 
til 25,  35,  45,  etc.,  are  reached.  The  general  order  of  procedure  will 
by  this  time  be  well  fixed,  aud  teachers  need  only  to  look  out  for  iudi- 
viilual  errors  such  as  putting  too  many  naughts  iu  the  quotient  when 
the  child  has  once  learned  to  jjut  any  there;  having  a  remainder  large 
enough  to  contain  the  divisor  once  more  aud  not  recognizing  the  fact, 
but  writing  a  one  in  the  quotient  before  continuing,  and  similar  errors. 
As  soon  as  the  process  is  mastered  so  that  accuracy  is  assured,  pupils 
may  work  for  speed,  both  in  diviiling  and  iu  checking  results." 
Minnesota  Course  of  Study  p.  112. 

SUBJECT  MATTER. 

NOTATION  AND  NUMERATION: 

Counting  to  100  by  15.  20.  and  25. 

■^riie  Koman  Numerals  L,  C.  au<l  D. 
ADDITION. 

Time  tests. 


55 

MULTIPLICATION. 

Tables  of  11  and   12. 

No  multipliers  over  four   places. 

Drill  for  speed. 

DIVISION. 

Loiiff  division,  using  a  two  place  divisor.     Zero  in  tin-  divisor. 
Drill  for  speed. 

DENOMINATE  NUMBERS. 

Kecognitiou  of  the  square  inch,  square  foot  and  scpiarc  yard. 
FRACTIONS. 

Part  taking, 

VI  and  X2  of  any  two  place  numbers. 

Addition  and    subtraction  of    fractions  wliose  denominators  may 

be  seen  by  inspection. 

DECIMALS. 

Addition  and  subtraction  of  V.  S.  money. 

Multplicatiou  and  division  of  U.  S.  money  by  integers  only. 

GENERAL  PROJECT. 

This  is  a  suggested  project  which  will  involve  much  of  the  sub- 
ject matter  of  the  grade.  Lest  it  shouhl  not  provide  sufficient  drill 
to  fix  the  processes,  minor  projects  are  given  as  details  under  each 
topic  of  subject  matter.  The  latter  are  unrelateil  but  might  be  an 
outgrowth  of  a  larger  project  similar  to  the  one   here  given. 

To  save  school  supplies.  This  project  may  be  used  to  involve 
arithmetic,  language,  spelling  and  industrial  arts.  The  jjroject  can 
be  followed  as  outlined.  Then,  when  need  arises,  the  concrete  work 
may  be  discontinued  for  a  week  or  so  to  give  necessary  drill  on  those 
processes  which  require  special  emi)hasis.  Since  thrift  in  these  t\i\ys 
of  high  prices  is  such  a  vital  problem,  and  since  directions  given  to 
the  whole  class  to  "Do  not  waste  your  i)aper,"  or  'Take  care  of  your 
pen-points,"  do  not  produce  the  desired  result,  graphic  representation 
of  the  expense  incurred  for  two  or  more  successive  months  is  likely 
to  decrease  waste,  since  it  is  a  concrete  problem  and  will  apj»eal  to 
most  of  the  class,  (rroup  infiuence  will  compel  those  who  are  not 
directly  reached,  to  l)e  more  careful.  Perhai)S  not  all  of  the  subject 
matter  for  the  IV-A  Grade  will  be  covered  by  this  i)roject,  but  most 
of  it  can  be  brought  in.  The  Roman  Numeral  1)  can  be  included  in 
connection  with  a  ream  of  pajicr  or  ')(»()  sheets;  C  as  a  number  of 
sheets  in  a  pad  of  drawing  i)ai)er.  The  table  of  I'i's  is  involved  in 
finding  the  number  of  articles  in  any  number  of  dozen  or  in  the  gross. 
Multiplication  and  long  division  can  easily  be  included.  The  pro]»- 
lems  given  are  merely  a  suggestion  as  to  how  the  minor  jiroject  may 
be  worked  out  more  definitely. 

LANGUAGE:  To  discover  who  pays  for  the  sui»plies  used.  (The 
School  IJoard  Init  indirectly  the  i)areuts.) 


56 


To  think  of  \vay.<  in  wliich  the  increased  cost  of  supplies  may  be 


tiiet. 


|>y  having  the  parent  pay  more  tax. 

liy  l>ein{:  more  careful  in  usinfr  supplies. 

Wliioh  way  is  the  best  ? 
The  foUowinir  i**  a  l»rire  list  of  the    articles  used  most  frequently 
in  the  schoolroom.  These  are   real  i)rices,  obtained  from  the  Account- 
in-:  Dej-artnieiit  of  the  Tx-anl    of  Education: 


ARTICLE 

PRICE 

Chalk 

7oc  per  gross 

2  pes.  for  Ic. 

Drawintr   Pajter 

4c  per  humlretl 

Ic  for  25  sheets. 

Desks 

S5.25  a  piece 

Erasers 

89c  per  dozen 

4c  a  piece. 

Tennianship  Hooks 

4c  a  piece. 

■/. 

18c  per  dozen 

2  cakes  for  3c. 

Ink  Wells 

SI. 60  per  dozen 

14c  a  piece 

Paper  Towelint: 

23c  a  roll 

5  sheets  for  Ic. 

Pen  Points 

46c  a  gross 

3  for  Ic, 

Pencils 

$2.25  per  gross 

2c  a  piece. 

Pencils  (drawin 

g) 

S3. 12  per  gross 

6  for  13c. 

Rulers 

lie  j)er  dozen 

lea  piece. 

Writing  Paper 

48c  a  ream 

10  sheets  for  Ic. 

Peaders: 

Eldsou  IV. 

45c.. 

Merrill  IV. 

56c. 

Natural  IV. 

56c. 

Reading  — Literature     56c. 

ARITHMETIC:  To  ligure  the  cost  of  the  writing  paper  used  in 
the  whole  school  for  month. 

How  many  sheets  of  writing  paper  were  used  l)y  the  whole  class 
in  penmanship,  this  week? 

How  many  weeks  are  there  in  a  school  mouth? 

How  many  sheets  wouhl  be  used  in  the  entire  month? 

How  many  children  are  in  the  room?    43. 

The  whole  room  uses  1324  sheets  in  a  month.  How  much  does 
one  child  use  ?    30. 

How  many  iiujjils  are  in  the  school? 

The  pui»ils  may  write  a  letter  to  each  room,  asking  for  the  en- 
r(dlment. 

If  each  child  in  the  school  uses  30  sheets  in  a  mouth,  how  much 
is  used  by  the  entire  school? 

4.395  boys  in  the  first  six  grades  of  the  Duluth  schools  use  iuk. 
How  many  sheets  of  pa])er  will  they  use  in  a  month? 

4,275  girls  will  use  how  many  sheets. 

How  many  sheets  <lo  both  the  l)oys  and  girls  use? 

To  find  the  total  value  of  the  supplies,  including  books  which 
each  child  has  in  its  desk. 

To  find  the  value  of  all  the  basal  reading  books  in  the  room;  all 
the  penmanshij)  >»ooklets;  penholders;  erasers;  rulers;  ink  wells;  desks. 


57 

etc.  From  these  results,  to  detertiiiiie  the  total  value  of  the  ix'nna- 
iieut  supjtlies. 

To  tififure  the  cost  of  sui)i)lie8  (coii3uiijal)le,such  as  i)eiicils,  pajier, 
peu-i)oiut)  ueeded  by  the  whole  I'ooni  iu  a  month. 

To  help  the  teacher  trraph  ou  the  blackboard  the  cost  of  the 
supplies. 


10'^ 

zo' 

3(f 

so' 

50* 

fed" 

70' 

80'- 

:?o" 

r 

X' 

2>^ 

i^ 

CHALK 

— ' 

— - 

-—^ , 

DRAWING-PAPER 

_^ 

^ 

WRITING-PAPER 

' 



— 

- . 

PENCILS 

__ 

— 

— 

— ' 

PEN-POINTS 

S 

PAPER  TOWELING 

\ 

ETC. 

\ 

LANGUAGE.  To  suyyest  methods  of  elimiijatiiif>-  waste;  i.e., 
writiug"  ou  both  sides  of  the  paper;  care  in  usin^  pen-])()iuts  and  iu 
dilipin<r  ink;  having^  clean  hands  when  usiujiif  books;  usiny  only  one 
pa])er  towel  at  a  time;  picking  up  chalk  instead  of  stepi)injj  upon  it, 
etc.     One  child  rniyht  be  made  responsible  for  each  of  the  sui)plies. 

To  take  part  in  a  campaign  to  see  how  much  less  the  expenses 
can  be  made  the  following  month. 

ARITHMETIC.  To  determine  the  saving  made  by  indivuluals 
and  by  the  whole  room  the  second  month. 

To  show  by  additions  to  the  graph,  the  saving  which  has  ensued. 

LANGUAGE.  To  tell  another  room  how  they  can  eliminate  waste. 
The  pupils  who  can  do  this  most  effectively  may  be  chosen. 


MINOR  PROJECTS. 

FUNDAMENTAL  PROCESSES. 

To  see  how  majiy  problems  can  be  done  in  one  minute,  A  num- 
ber of  addition  or  subtraction  problems  are  i)laced  on  the  board.  The 
children  test  their  rapitlity  and  accuracy  by  seeing  how  many  of 
these  they  can  solve  correctly  in  one  minute,  the  teacher  giving  the 
signals  to  begin  and  stop.  The  individual  scores  are  kept  on  the 
board  and  these  should  serve  as  a  stimulus  toward  improvement. 

To  see  how  long  it  takes  to  do  five  a<ldition  and  subtraction 
problems. 

This  test  is  carrietl  on  in  the  same  way,  the  teailicr  timing  the 
children  as  to  ra])idity.  Later  they  are  scored  as  to  accuracy  ol  their 
Work,  these  scores  also  are   kejtt  on  the  board. 

To  play  Hlackboard  Uelay.  Two  sets  of  exami)les  arc  place.l 
on  the  board.  The  class  is  di-vi<led.    The  leader  iu  each  ilivision  writes 


58 

the  answer  to  the  first  examples,  returns  tlie  c-lialk  to  the  next  impil 
who  works  the  next  example  aud  so  on  until  all  have  had  a  chance. 
There  shouhl  be  as  many  examj^les  as  children.  The  wiuning-  side  is 
the  one  whicli  first  conijiletes  the  examples  correctly.  A  child  may 
<-orrect  any  errors  in  work  done  i)reviouslj'.  Decatur  Course  of  Study. 
T.i  see  who  can  first  fill  out  his  card.  ]Make  cards  with  fiijures 
from  1  to  12  alony  the  to])  and  left  liand  maryin;  the  remainintj 
si)aces.  makintr  up  the  twelve  tables,  heiny  left  blank.  Proviile  clieck- 
ei"s  with  tlie  necessary  multiples  irom  1  to  144.  Eacii  checker  has  a 
])lace  in  one  of  the  blank  si>ace.s  on  the  card.  When  i)roperly  placed 
tlie  tables  froiif  1  to  \'2  are  complete.  The  one  rilliny  liis  card  first  is 
winner.    Mankato  Course  of  Study. 

DENOMINATE  NUMBERS. 

To  cut  sipiare  inches  and  square  feet  out  of  paper  and  use  them 
in  covering  the  surfaces  of  books,  desks  and  i)apers. 

To  estimate  the  area  of  books,  desks,  window  panes,  etc.,  verify- 
ing- tlie  same  by  actual  measure,  seeini>-  who  can  estimate  most  accur- 
ately. 

To  find  out  which  blackboard  is  longest;  how  long:  the  sand  table 
is;  how  wide  it  is;  how  many  square  feet  it  contains.  How  long  and 
wide  the  teacher's  desk  is.  How  long  and  wide  the  i)upirs  desk  is. 
Ildw  long  and  wide  the  school  room  is. 

To  solve  problems  relating-  to  the  lumber  industry.  Facts  which 
might  be  used: 

UNITED  STATES  Millions  of  M  Feet 

Washington 4,592,058 

Louisiana 4,1()1,56U 

Oregon 2,098,4G7 

Mississippi 2,010,581 

Texas 2,081,471 

No.  Carolina 1,957,258 

Arkansas 1,911,647 

Alabama 1,528.986 

Wisconsin 1,498,858 

\'irginia 1,278,958 

AV.  Virginia 1,249,559 

Michigan 1,222,988 

California 1,188,880 

Minnesota 1,149,704 

Florida 1,055,047 

All  others 8,822,021 

None:  These  numbers  may  be  i)ut  in  more  sim])le  form  if  their 
length  confuses  the  class. 

To  find  the  total  lumber  pro<luctioii  of  the  Cnited  States;  differ- 
t'lice  in  jtroduction  lietween  states. 

To  find  how  much  lumber,  iron,  grain,  is  produced  by  the  lead- 
ing countries;  difference  in  i)roduction  between  countries  or  states. 


59 

WHEAT  (In  Bushels) 

COUNTRIES. 

Russia (J99,4I8,()()() 

I'^uited  States. __   _    t)9;'),44:{,0(»(l 

ludia 857,941,0(10 

France 259,1X0.000 

Austria-Hung-ary 241,:}i»4.000 

Italy 158,:i:{7,000 

Canada 149,990,000 

Spain '__ ^^___1B7.44S,000 

STATES. 

Kansas 177,200.000 

No.  Dakota SI, 592, 000 

Nel)raska 6S,1  ]  (5,000 

Oklahoma ._  47,975,000 

Missouri 43,888,000 

Indiana_______ ___':____  48,289,000 

Minnesota 42,975,000 

Washiufiton 41,840,000 

Ohio _. 36,58S.O0O 

South  Dakota. 31,5()().000 

All  others 276,(548,000 

Finding  the  total  production  of  the  V.  8.;  difference  in   produc 
tion  of  various  states. 

IKON  ORE  (By  Tons-1910) 

COUNTRIES. 

United  States 26,10S,199 

(Terraany .    1 2,91 7, (J58 

United  i{inffdom___ ___.   9.81S,91G 

Fran  ce 3,682. 1 05 

Russia 2,871.882 

Austria-Hung-ary 1,958.786 

Belgium 1,682,850 

Canada 687,928 

STATES  (In  millk  ns  of  dollars) 

Pennsylvania 688.922.000 

Ohio 281.479,000 

Illinois 124,908,000 

New  York 66,158,000 

Alabama 21,286,000 

All  others 214,454,000 

Standards  of  Attainment. 

WOODY  TEST.  Series  A. 

Addition ^^-'^ 

Suhtraction l-i-'> 

Multiplication •'•' 

Division 9.8 

COURTIS  TEST:  Series  B -Speed 

Addition |^-'* 

Subtraction '  •" 

.Multiplication '»•" 

Division "*•" 


(30 

COURTIS  TEST  Series  B. -Accuracy. 

AtUlitiou 64. 

Siilitnu-tioii ^^0. 

M  ult  i  plicat  iou 67. 

Division 57. 


Bibliography. 

TEACHER'S  READING.      1)1h>\\  u    aud    Coftniau:    How  to  Teach 

Arithmetic. 

C'hai).  III.        Accuracy. 

Chap.  V.  Markiuy  papers. 

Pp.  cS,S-91.      Analysis. 

Chap.  VIII.    Value  of  drill. 

Chap.  XI.       Teaching  the  fuudanieutals. 
Kendall  and  Mirick:  How  to  Teach  the  P^undameutal  Subjects. 

Pp.l7U-19U.  Skill  in  Calculation. 
Skill  in  Application. 
Inductive  Teaching. 
Mental  and  Oral  Lessons. 

Pp.  195-200.    Drills. 

Tests  and  Katings. 
Klapper:  The  Teaching  of  Arithmetic. 

Pp. 48-51.        Arithmetic  Must  be  Humanized. 
Motivation  in  Arithmetic. 

Chai).  IV.       General  Principles. 

Chap.  VIII,   Multi])lication  and  Division. 
McMurry:  Special  Method  in  Arithmetic. 

Chap.  IV.       Method  for  Intermediate  Classes. 
Smith:  The  Teaching  of  Arithmetic. 

Chap.  IV.       The  Nature  of  Problems. 

Chap.  IX.       Children's  Analyses. 

Chap.  X.         Interest  and  P]ffort. 

Chajj.  XII.     Principles  of  Teaching  Arithmetic. 

Cha}).  XVIII.  The  Fourth  School  Year. 
Strayer  and  Norswortliy:  How  to  Teach, 
"p.     204.         The  Drill  Lesson. 

l*p.  2H4-248.  Measuring  Achievement. 
Suzzallo:     The  Teaching  of  Primary  Arithmetic. 

Chap.  \'III.  Methods  of  Pationalization. 

Chap.  IX.       Special  Methods   for   Obtaining  Accuracy  and 
Speed. 
Wilson:  The  Motivation  of  School  Work. 

Chap.  IX.       Motivation  of  Arithmetic. 

SUPPLL.ME.NTARY  BOOKS. 

Hoyt  and  Peet:  Everyday  Arithmetic,  Book  1. 

Stone  and  Millis:   Primary  Arithmetic. 

Thorndike:   Arithmetic,   IJook  1. 

Wentworth  and  Smith:  Essentials  of  Arithmetic,  Primary  Book. 


i 


61 
GRADK  V-i;. 


DIRECTIONS. 


DIVISION  OF  TIME.  Oiie-half  of  the  work  of  tliis  <rraile  should 
he  oral  and  one-fourth  should  be  concrete. 

REVIEW.  Fundamentals,  showing-  the  relation  to  fractions, 
tables  of  ll's  and  12's,  lony  divison  with  two  j)lace  divisor. 

MAIN  TOPICS:  Lon};  division  with  three  i)lace  divisors,  re<luc- 
tion,  addition,  and  subtraction  of  fractions,  and  retluction  of  a  few 
fractions  to  decimals. 

ADDITION,  SUBTRACTION,  AND  MULTIPLICATION.  The  drills 
jfiven  on  these  topics  should  be  for  speed  as  well  as  accuracy. 

DIVISION.  Increase  the  divisor  to  three  places  but  no  divisor 
should  exceed  three  places. 

The  special  work  of  this   {>rade  is   ailditioii  and   subtraction   of 
fractions.     This  involves  a  study  of  the  followinfi-  whicdi  must  be   un- 
derstood before  certain  phases  of  the  work  can  be  taken  up: 
Factoring. 
Multiples. 

Tests  for  divisibilities. 

Least  common  multiple  of  numbers  commonly  found  as  de- 
nominators. 

Reduction  of  improper  fractions  to  mixed   numbers   ami  the 

reverse. 

Use  such  fractions  as  conform  to  business  practices  and  show 
that  the  denominator  is  only  the  name  of  the  i)art  taken. 

LEAST  COMMON  MULTIPLE.  "Develop  the  L.C.  M.  only  as  much 
as  required  in  tindinfj  the  least  common  denominator.  It  is  really  doubt- 
ful if  addition  and  subtraction  of  fractions  shouM  extend  to  fractions 
too  largfe  to  i>ermit  pupils  to  determine  the  common  denominator,  by 
inspection'".     Connersville  Course  of  Stu<ly  in  Mathematics. 

"Much  valual)le  time  is  often  lost  because  teachers  interpret  the 
expression  "by  insi)ection"  to  mean  "without  any  computation  what- 
ever", or  "by  guessing".  "By  inspection"  must  be  interpreted  to 
mean  "without  written  computation";  hence,  children  must  be  taught 
a  simi>le  means  of  determining-  the  least  common  denominator". 
Klapi)er. 

The  following  is  suggested  as  jjerhaps  the  easiest  for  the  children: 

"Take  the  first  two  numbers  and  find  L.  C.  ]M.,  then  tind  the  L. 
C.  M.  of  that  and  the  next  figure  4,  ti,  9;  L.  ('.  M.  of  4  an.l  (i  is  \'2:  L. 
C.  M.  of  12  and  9  is  3(5."     Klapper. 

ADDITION  AND  SUBTRACTION  OF  FRACTIONS.  These  may  \>v 
taught  simultaneously,  the  order  of  difficulty  being  the  same: 

Similar  fractions. 

Dissimilar  fractions. 

Mixed   numbers. 

Simple  two-ste])  problems  may  now  be  useil  but  recpiire  no  more 
than  two  daily.  Lead  the  children  to  decide  upon  the  processes  nec- 
essary before  beginning  to  work. 


62 
SUBJECT  MATTER. 

DIVISION.    Three  place  divisors. 

FRACTIONS.     Need  of  fractions. 

Keiiiu-tioii  to  hifj-her  terms;  to  lower  terms,  to  au  integer  or  a 
inixe<l  number;  to  a  common  denominator;  to  like  denominators. 

Addition  and  sul»trac-tion  of  similar  fractions;  dissimilar  frac- 
tions; mixeil  nurn1»ers;  application,  ad<lition  and  subtraction  problems 
of  child  experience. 

MEASURES.     !\linutes  in  the  hour  and  seconds  in  the  minute. 

DECIMALS.  In  the  reduction  of  fractions  all  fractions  with  10, 
lUO,  or  lUUO  as  denominators  may  be  expressed  as  decimals.  Deci- 
mal i)oint  in  V.  S.  money. 

OPTIONAL  WORK.    Divisors  of  more  than  three  places. 
Adilition  and  subtraction  of  larger  fractions  and  mixed  numbers. 
Extcnde<l  drill  on    V.  S.  money. 

GENERAL  PROJECT. 

This  is  a  suyg'estefl  project  which  will  involve  much  of  tlie  sub- 
ject matter  of  the  tjrade.  Lest  it  should  not  provide  sufficient  drill 
to  tix  the  i)rocesse8,  minor  projects  are  given.  Tlie  latter  are  unre- 
lated, but  might  be  an  outgrt)wth  of  a  larger  project  similar  to  the 
one  here  given. 

To  Plan  a  Camping  Trip.  The  class  should  deci<le  when  and 
where  to  go  and  make  the  general  outline  of  their  plans. 

FRACTIONS  AND  U.  S.  MONEY  AND  MEASUREMENTS.  What 
will  be  the  best  way  to  go,  by  train,  by  automobile  or  on  foot? 

If  an  automobile  travels  1  mile  in  Bi  min.,  -what  would  it  cost  to 
make  the  trij)  by  automobile  at  60c  per  hour  for  a  five  passenger  ear? 

If  you  are  delayed  by  a  i)unctured  tire  1 1  hours,  by  a  broken 
axle  2}  hours,  by  a  blow-out  l^'s  of  an  hour,  and  by  minor  accidents 
^'s  of  an  hour,  «hat  is  the  entire  delay  on  yotir  trip?  How  long, 
then,  would  the  trij)  take? 

Figure  the  cost  of  gasoline  at  24c  per  gallon  if  some  member  of 
the  class  has  an  automobile  that  could  be  used. 

What  would  the  railroad  fare  be  at  8c  per  mile  with  a  war  tax 
of  Ic? 

How  long  would  it  take  to  walk  if  the  jiarty  walked  seven  hours 
daily  at  tlie  rate  of  ]  of  a  mile  in  ten  minutes? 

If  you  walk  6^  miles  the  first  day,  o!,*  the  second,  7]  the  third, 
6|  the  fourth,  how  far  have  you  walked  in  4  days?  How  far  are  you 
still  from  your  destination?  What  was  your  average  rate  i)er  day? 
At  that  rate  how  long  woidd  it  take   you  to  complete  the  trij)  ? 

What  car  must  you  take  to  get  an  N.  P.  train  that  leaves  the 
rniiin  Station  at  9:10  a.m.? 

\i  the  train  travels  2.S  miles  per  hour,  at  what  time  will  you  reach 
your  destination? 


63 

]\[ake  a  list  of  piolialilc  ueeessities  in  tlic  way  of  food  and  nilier 
supplies.  What  food  sui)plies  must  be  taken  in  order  to  have  fii(Mi<rl, 
for  1  week':'     For  the  entire  trii)?     What  will  l)e  the  (-(.st  ? 

If  eaeh  i)ers()n  uses  4  lb.  of  butter  <laily.  Iiow  inudi  niiist  lie  taken 
to  supply  the  entire  class  for  one  week? 

A  soldier  was  o-iveu  2  oz.  of  butter  daily.  On  arni\-  rations,  how 
niueli  butter  would  be  needed  for  one  week?     For  the  entire  trip? 

If  2  lbs.  of  butter  are  used  daily  for  cookijiy:,  how  rnueh  money 
would  be  saved  in  a  week  by  usiiiy-  butter  substitutes  if  1  lb  of  olco- 
inar-iarine,  \  lb.  of  lard,  and  l  lb.  of  suet  take  the  place  of  1  11>.  of 
butter?    lIow  niiudi  would  be  saved  duriiify  the  entire  ti-ip? 

What  will  be  the  cost  of  3  hams  weiyhinji-  respectively  \'1\  lbs., 
lOs  lbs.,  and  II f  lbs.,  at  25c  ])er  pound. 

If  4  cupful  of  Hour  makes  4  <iriddle  cakes,  how  many  cui)S  of 
Hour  will  be  retpiired  to  nuike  enoujih  cakes  for  all,  allowing  (5  cakes 
for  each  person. 

The  following  recipes  are  intended  for  6  persons: 
To  make  apple  sauce,   'l  cupful  of  sugar  to  (j  or  8  apples." 
Lemouade,  "Lemon  .juice,  l  cupful;   water,  1  qt.;  sugar,  1  cui>ful. 
Adapt  these  recipes  to    make    enough  for  one  serving  for  all  an(i 
estimate  (piantity  of  each  ingredient  needed  and  cost. 

LIQUID  MEASURE.  If  each  person  uses  I  pt.  of  milk  daily,  how 
much  will  be  needed  each  day?     What  will  it  cost  at  17c  a  (juart? 

The  food  value  of  one  quart  of  milk  is  about  the  same  as  that  of 
9  ounces  of  round  steak,  or  (S  eggs.  Look  up  prices  and  see  which  is 
the  most  econoTuical  and  how"  much    difterence  there  is. 

Let  class  choose  a  committee  to  plan  some  athletic  games  and 
contests  for  the  trip  which  may  be  practiced  in  the  school  room  or  on 
school  ground  such  as  the  hammer  throw: 

Use  cardboai'd  as  paper  hammer.  ^Measure  accurately  eacli  con- 
testant's distance  to  fraction  of  an  inch.  Determine  the  wiinur  by 
fraction  of  an  inch. 

IIow'  long  would  it  take  >"ou  to  earn  and  save  enough  money  to 
go  on  this  tri])? 

If  you  earn  4  of  a  dollar  every  week  and  si)end  \  of  a  dollar  how 
long  would  it  take  you  to  save  one  dollai?  How  long  to  earn  suHi- 
cient  money  for  the  trip? 

Make  out  a  bill  for  the  necessary  food  supplies   for  1  day. 

Let  pupils  keep  itemized  account  of  pe.'sonal  exi)enditures  and 
tinally  include  in  this  their  share  of  total  expenses. 

MINOR  PROJECTS. 

DIVISION.  To  tinil  the  amount  of  land  each  child  will  have  tor 
a  school  garden.  There  are  5400  S(i.  ft.  of  land  besidi's  the  land  for 
paths.  There  254  pupils  in  the  school  but  45  have  gardens  at  home. 
IIow  many  sq.  ft.  will  each  i)Upil  have,  w  ho  makes  his  garden  at 
school? 


64 

FRACTIONS.     To  fiinl  out  how  imu-h  siigrar  I  shall  nee<l. 

I  j>laii  to  make  a  cake  retiuiriiiii-  \  lb.,  another  reciuirinjuf  5  1V»..  a 
pie  re(juiriujf  5  11>..  aii<l  .>Juiiie  cookies  requiriug  5  lb.  How  niuchsutfar 
must  I  have? 

ON  TMK  ATHLKTIC  FIELD. 

To  timl  liow  many  seconds  it  takes  a  ball  player  to  make  a  home 
run,  it*  it  takes  him  4 5  seconds  to  set  to  first  base,  A\  seconds  to  get  to 
second  ])ase.  4|  seconds  to  get  to  third  base,  and  45  seconds  to  get  to 
the  home  plate. 

To  find  how  many  seconds  over  a  quarter  of  a  minute  is  the 
time  taken  for  the  home  run  in  problem  1. 

To  find  the  difference  in  the  length  of  the  running  broad  jumps 
of  Albert  whose  record  is  12 1  feet  and  Charles  whose  record  is  14 1% 
feet. 

To  timl  the  <litlereuce  in  time  between  a  record  of  4f  minutes 
for  a  mile  run  and  one  of  bj  minutes. 

To  fin<l  the  difference  in  length  of  a  running  track  i\  of  a  mile 
long  and  J  of  a  mile  long. 

IN  CAMP. 

To  find  the  increase  in  a  boy  scout's  walking  record  if  on  one 
tramp  he  extends  his  record  of  H\  miles  to  one  of  14  miles. 

To  find  the  distance  covered  in  a  day  by  a  group  of  Camp  Fire 
girls  who  walk  0}  miles  in  the  morning  and  4^  miles  in  the  afternoon. 

To  find  how  far  two  boys  are  from  the  end  of  a  trail  that  is  6^ 
miles  long,  after  having  climbed  4jmiles. 

To  find  how  far  from  the  farther  shore  is  a  canoe  which  has  been 
paddled  Ij  miles,  if  the  lake  is  'If  miles  across. 

To  find  the  difference  in  time  in  a  hundred  yard  swimming  con- 
test, between  a  record  of  22|  seconds  and  a  record  of  1  minute. 

To  find  the  difference  in  a  walking  contest,  the  winner  having 
walked  a  mile  in  S^^  minutes, the  next  boy  in  the  contest  took  10]  min- 
utes. 

To  find  the  number  of  cups  of  sugar  Dora  must  borrow.  Dora 
is  making  jelly  and  needs  BO  cup  of  sugar  but  she  has  only  24^^  cups. 
How  many  cups  must  she  borrow  from  a  neighbor  to  finish  her  jelly':' 

MEASURES.  See  second  iirojcct  under  Fractions. 

To  find  out  what  time  the  street  car  leaves.  Beginning  at  S:.50  a 
car  passes  the  house  every  20  minutes.  At  what  time  after  9:00a.  m. 
can  1  catch  a  car? 

To  finil  out  what  tiiiu'  I  must  leave  the  house.  If  it  takes  5  min- 
utes to  walk  to  the  corner,  wlicn  must  I  leave  the  house  to  catch  a 
car  about  9:30? 

To  find  out  when  I  shall  arrive  down  town.  It  takes  23  minutes 
to  go  down  town.  H  I  take  the  9:50  car,  when  will  I  arrive  down 
town? 


i 


65 

DECIMALS.  To  order  some  l)()oks.  (Tive  the  eliiM  a  prit-e  list  of 
some  child reu's  books  as: 

The  Juuffle  Book  §1.50. 
Niffhts  with  Uncle  Keinus  >?1.^0. 
Kobiuson  Crusoe  $.75. 
Alice  ill  Wonderland  n.45. 
Grimm's  Fairy  Tales  >;.45. 
Arabian  NiiJ^lits  >!.50. 
Hans  Brinker  -s.50. 
The  Dutch  Twins  ^M. 

Fees  for  money  orders.    For  orders  from: 

§.01  to  §2.50 3  cents 

S2.51  to  §5.00 5  cents 

§5.01  to  §10.00 S  cents 

§10.01  to  §20.00 10  cents 

Select  several  books  you  would  like  to  have  and  find  the  cost  of 
the  order  allowinof  for  the  cost  of  the  money  order.  Vary  this  by 
ordering  seeds,  toys,  plants,  tools,  etc. 

Standards  of  Attainment. 

Fundamentals,  see  V-A. 

Reasoning-:  The  child  should  have  ability  to  use  each  new  acquisi- 
tion until  he  has  control  of  it;  he  should  have  a  working  knowledge 
of  the  needed  processes. 

OBJECTIVE.     See  Grade  V-A. 

Bibliography. 

METHOD.     Brown  and  Coffmau:   ITow  to  Teach  Arithmetic. 

Ghap.  XIII,  Common  Fractions. 
Kla])i)er:  The  Teaching  of  Arithmetic. 

Chap.  X,  Pp.  21S-242. 
Wilson  and  Wilson:   The  .Motivation  of  school  work. 

Chap.  IX,  The  Motivation  of  Arithmetic. 

SUPPLEMENTARY  BOOKS.  IToyt  and  Peet:  Fveryday  Arith- 
metic. Book  Two. 

Part  III,  Chap.  Ill,  Addition  and  Subtraction  of   Fractions. 
Part  IV,  Chap.  IV,  Addition  and  Subtraction  of  Fractions. 

Stone  &  Millis:  The  Stone-Millis  Intermediate  Arithmetic. 
Chap.  Ill,  Pp.  79-97.  Fractions. 

Thorndike:  The  Thormlike  Arithmetic,  Book  Two. 

Wentworth  and  Smith:  Essentials  of  Arithmetic.  Intermediate 
Book,  Chap.  II.  Pp.  47-7(1.  Addition  and  Subtraction  of 
Fractions. 


(;1{A1)K  V-A 


DIRECTIONS. 


I)l\  ISION  (JF  TIMi:.  Oue-balf  of  tlu'  aritliinetic  tijiie  allotineiit 
of  this  yraile  should  l»e  devoted  to  oral  work. 

One-fourth  of  the  time  should  he  devoted  to  concrete  ])rol)lems. 

KE\'IE\V.  Keductiou,  addition  and  subtraction  of  fractions;  U. 
S.  money;  drill  on  the  fundamentals  for  speed  and  accuracy;  the 
tables  of  denominate  numbers  i)revii>usly   learned;  decimals. 

MAIX  TOPICS.  .Alultiplication  and  division  of  fractions;  tables 
of  linear,  dry,  and  liquiii  measure,  weij^ht,  time  and  U.  S.  money. 

FRACTIONS.  In  teaching  multiplication  and  its  reverse  pro- 
cess, division,  the  procedure  should  be  (l)  multiplyinjr  or  dividing-  a 
fraction  by  a  whole  nucnber,  2  (a)  whole  number  by  a  fraction,  (8)  a 
fraction  by  a  fraction,  (4)  a  mixed  number  by  a  whole  number,  (5) 
a  mixe  1  numVjer  by  a  mixed  number. 

There  is  no  good  reason  why  a  child  should  remember  any  of  the 
explanations  of  the  processes  in  fractious;  it  is  sutticient  that  he  learn 
tlie  operation  as  a  rational  one,  and  that  .he  can  perform  it  quickly 
and  accurately.  What  we  want  is  control  of  fractions,  power  to 
work  with  them,  whether  with  or  without  analytic  understanding. 
Use  cancellation  wherever  possible  after  the  processes  have  been 
mastered.  Mixed  numbers  should  conform  to  business  ])ractice,  64, 
12.';,  (S;^,  etc.     There  should  be  much  mechauical  drill. 

DENOMINATE  NUMBERS.  Much  of  the  work  in  denominate 
numbers  should  be  oral  and  should  be  related  to  the  daily  transac- 
tions of  business  life.  Problems  involving  reductions  through  more 
than  three  denominations  are  seldom  used  in  the  business  world.  As 
far  as  possible  the  children  should  have  the  actual  measures  present- 
eil  to  their  senses.  It  is  easy  to  make  tlie  mistake  of  talking  abouJ 
these  things  without  children  actually  having  a  kjiowledge  of  them. 
Thor«tughly  memorize  the  tables. 

-AI.KJUOT  PARTS.  Teach  the  common  aliquot  parts  of  a  hun- 
dred with  their  fractional  equivalents.  These  may  be  used  as  time 
savers  in   multiplication. 

SUBJECT  MATTER. 

FRACTIONS.     Multiplication,  division,  (  anrcllation. 

DENOMINATE  NUMBERS,  involving  tables  of  linear,  dry  and 
liquid   measure,  weight,  tiine  and  U.  S.  money, 

ALIQUOT  PARTS.  Those  most  commonly  used  .12.^,  .25,  .50,  .75, 
.20,  .10,  with  their  fractional  c(iuivalents. 

OPTIONAL  WORK.  Work  for  liigher  degree  of  speed  and  acc-ur- 
acy.   Solve  more  dithcult    iimblcms. 

GENERAL  PROJECT. 

This  is  a  suggested  project  which  w  ill  involve  much  of  the  sub- 
j<'ct  matter  of  tiie  giade.      Lest  it  sliould    not    proviile    sufficient   drill 


67 

to  fix  the  i)rocesseH,  minor  iirojects  are  tfiveii.  The  hitlci-  ui-c  unrc- 
hiled  but  Miitiht  be  an  oiittiTowtli  of  a  lary-er  jiroject  similar  to  (lie  one 
here  given. 

To  Plan  a  Recreation  Park  on  a  City  Block  in  Duluth. 

LINEAR  MEASURE  AND  FRACTIONS. 

Make  a  scaled  (lra\vin<4-  of  tlie  lilock. 

I'ut  in  the  drawinii'  of  the  block  a  ball  diamond,  a  tennis  court 
or  anything  upon  which  the  class  may  decide.  Make  measurements 
and  calculations  accurately  to  the  fraction  of  an  inch. 

In  case  the  class  decides  ui)on  a  base  ball  diamond  ami  a  iciuiis 
court,  the  following  problems  may  grow  out  of  the  project: 

If  9  board  feet  will  make  1  yard  of  fence  1  board  high,  iiow 
many  board  feet  of  lumber  will  be  reiiuired  to  put  a  fence  4  boards 
high  around  the  entire  block";' 

How  much  will  the  fence  cost  at  §B0.00  per  thousand  board  feet? 

How  long  a  pole  will  be  required  to  cut  8  posts  if  each  jiost  is 
4s  feet  long':* 

How  many  such  i)oles  will  be  required  for  the  i)osts  if  the  posts 
are  set  6  feet  apart? 

If  a  club  of  12  meml)ers  is  paying  the  exi)en8e  what  is  each  mem- 
ber's share  of  the  expense  of  the  fence  around  the  block? 

Find  the  cost  of  putting  a  high  wire  fence  around  the  tenuis 
court  at  40c  per  yd.    What  is  each  member's  share  of  tins  expense. 

\^'EIGHT,  TIME,  U.S.  MONEY,  LIQUID  MEASURE. 

Find  the  number  of  pounds  of  rock  necessary  for  putting  '1  two- 
inch  layers  of  rock  on  the  base  ball  diamond  if  it  takes  275  tons  for  1 
layer. 

How  much  will  be  used  in  the  tennis  court  for  2  layi'rs  if  the 
court  is  i  as  large  as  the  base  ball  diamond. 

How  many  trips  will  the  truck  driver  need  to  make  to  deliver 
the  rock  from  the  nearest  crusher  if  he  can  take  3  tons  per  load? 

How  many  hours  will  he  work  if  he  travels  at  the  rate  of  10 
miles  ])vv  hour,  allowing  15  minutes  foi-  loading  and  unloading  catdi 
loa<l  V 

What  will  it  cost  to  have  the  rock  delivered  if  the  driver  charges 
>;8.;")0  per  hour? 

What  will  be  the  cost  of  the  rock  for  base  ball  diam..nd  ancl  ten- 
nis c(jurt  if  the  first  layer  costs  '^1.2o  i)er  ton.  the  second  -sl.oO  per 
ton,  and  a  third  layer  consisting  of  1(10  tons  of  tine  rock  costing  oOc 
l)er  ton?  What  will  be  the  cost  of  spreading  if  labor  costs  .')0c  per 
hour  and  oTie  man  can  spread  2  tons  in  1  hour? 

In  making  the  court  and  diamond,  water  is  nee<led  in  i>utting  on 
each  layer  of  rock.  If  it  requires  133;'  gallons  for  each  layer  for  the 
diamond,  how  much  will  be  re(iuire<l  for  tlie  three  layers? 

How  much  will  lie  refpiired  for  the  tennis  court  if  ii  '-  '  :•> 
large  ? 

How  iiiaiiN    liairels  of  water  will  be  necessary  for  both? 


68 

After  the  water  is  ]»ut  on.  each  layer  of  rock  is  rolled  several 
times  hy  a  roller  drawn  liy  horses. 

How  lonjj  will  it  take  a  man  to  roll  the  three  layers  of  the  tenuis 
court  if  he  can  rt)ll  1  layer  iu  40  min? 

How  lontr  will  it  take  to  roll  the  base  ball  diamond  which  is  twice 
as  lar^re  ? 

What  will  be  the  entire  cost  of  rollinfj  for  both  if  he  charjo^es 
>;1.1*2  i)er  hour? 

Make  a  pay-roll,  keepinjf  the  time  of  the  labor  necessary  for  get- 
ting: the  rock  from  the  crusher,  spreading  and  rolling  both  the  base 
l»all  diamond  and  tennis  court. 

What  will  be  the  entire  expense  of  putting  rock  on  both? 

What  is  each  club  member's  share  of  this  expense  ? 

The  above  i)roblems  are  suggestive  only.  Many  others  will  grow 
out  of  the  project  and  l)oth  the  project  and  figures  may  be  changed  to 
suit  the  needs  of  the  class. 

MINOR  PROJECTS. 

FRACTIONS-  To  find  the  number  of  yards  of  matting  required 
for  a  ]>edroom  fioor,  if  it  requires  3  strips  4^  yds.  long. 

To  find  the  number  of  strawberry  plants  in  a  row  and  the  total 
number  of  plants  required  if  the  bed  is  1H\  ft.  long  and  8^  ft.  wide, 
and  l-l  ft.  is  to  be  allowed  for  each  plant.     Rows  run  lengthwise. 

To  draw  a  plan  of  the  strawberry  bed  to  scale. 

To  draw  to  scale  of  1  ft.  to  |  of  an  inch,  a  rectangle  to  represent 
a  fiower  betl  8  ft.  wide  and  16  ft.  long. 

To  find  the  value  of  24  yards  of  dish  toweling.  The  sewing 
teacher  charged  a  girl  24c  for  a  dish  towel  4  of  a  yard  long.  How 
much  did  the  sewing  teacher  pay  for  the  24  yds.  of  the  cloth? 

DP:.\0.M1XATE  numbers.  To  fiu<l  what  commodities  are  sold 
Ijy  the  pound,  the  dozen,  the  pint,  the  (luart,  the  gallon,  the  peck,  the 
bushel  and  the  ton. 

To  find  the  cost  of  the  number  of  feet  of  wire  required  to  fence 
a  jjlayground. 

To  find  the  difference  in  paying  room  rent  by  the  week  or  by 
the  month.  If  you  had  a  room  to  rent  for  a  year  which  would  you 
rather  have  §2.r)0  i>er  week  or  §10.00  a  month? 

To  find  the  number  of  number  of  half-pint  bottles  required  to 
bottle  3'  gallons  of  cream. 

To  find  how  wide  ycm  would  you  have  to  cut  a  ruffle  iu  order  to 
have  it  84  iu.  wide  when  finished,  if  I  '  in.  is  turned  under  on  one 
side  and  4  iu.  on  the  other  side. 

To  find  the  cost  of  4  oz.  of  ciuuamom  if  1  lb.  costs  40c. 

To  find  how  much  money  Henry  had  left  out  of  a  dollar  if  he 
bought  6  pencils  at  2  for  .5c,  a  pencil  box  for  25c  and  a  set  of  draw- 
ing tools  for  40c. 

To  make  an  itemized  acccniut  of  your  mother's  expenses  for  a 
week. 

To  keep  an  account  of  the  time  spent  at  some  useful  work  dur- 
ing the  week. 


69 
Standards  of  Attainment: 

The  child  should  he  able  to  fultill  the  re(iuireiiieiits  indicated  ])e- 
low: 

Deterniiue  all  ])roces8es  before  proceediuy  to  solve  hi.s  prohlein.s. 

Know  how  to  check  liis  results. 

Ilantlle  liis  processes  iu  whole  numbers  and   fractions   accurati-ly 
and  with  reasonable  speed. 

Have  memorized  his  denominate  number  tables  thorou<ihly. 

Read  and  write  decimals  through  thousandths  rapidly. 

Express  himself  freely  in  arithmetical  lauy-uage. 

Objective. 

Woody  Test:  Series  A. 

Addition 23.1 

Subtraction 20.4 

]\1  ultiplication 18.3 

Division 16.4 

Courtis  Test.  Series  B.  Speed. 

Addition 8.0 

Subtraction 9.0 

Multiplication 8,0 

Division 6.0 

Courtis  Test.  Series  B.  Accuracy. 

Addition 70 

Subtraction S3 

Multiplication 75 

Division 77 

Stone  Ueasouing  Test: 

Actual  Medians  Obtained. 

Butte,  1914 2.2 

Bridgeport,  Conn.,  1913 6.1 

Salt  Lake  City,  1915 3.7 

Lead,  S.  Dak.,  1916 4.5 

Tentative  standards  suggested  by  Stone   (1916)    for  his  reason- 
ing test — 

That  80  per  cent  or  more  of  5th  grade  pupils   reach  or   exceed   a 
score  of  5.5  with  at  least  75  per  cent  accuracy. 


Bibliography: 

TEACHER'S  READING.     Drown    and    Coftnian:     How    to    Teach 
.Vrithnietif. 

C'haii.  XII.  Denominate  Numbers, 
("liaji.  XIII,  Common  Fractions. 

Klai>iier:  The  Teaching  of  Arithmetic. 

Pp.  '2o2-241.  Multiplication  and  Division  of  Fractions. 

Wilson:  Tlie  Motivation  of  School  \V(»rk. 
Chap.  IX.  [Motivation  of  Arithmetic. 

SUPPLEMENTARY  BOOKS.     Iloyt    and    Peet:    Everyday   Arith- 
metic, Book  II. 

Part  Three,  Chap.  IV,  Multiplication  and  Division  of  Frac- 
tious. 

Part  Four,  Pp.  55-()3.  Multi])lication  and   Division   of  Frac- 
tions. 

Part  Three,  Chajt.  VIII,  Denominate  Numbers. 
Part  Four,  Chap.  VII,  Denominate  Numbers. 

Stone  and  Millis:  Intermediate  Arithmetic. 
Chap,  y,  Fractions. 
Pp.  97-125,  Multiplication  and  Divisou  of  Fractions. 

Thorudike  Arithmetic,  Book  Two. 

Pp.  49-55,  Multiplication  and  Division  of  Fractious. 
Pp.  90-122,  Denominate  Numbers. 

Wentworth  and  Siuith:  Essentials  of  Arithmetic. 
Pi).  ()S-97.  Fractious. 
Chap.  III.  Denominate  Numbers. 


71 

GIJADK  Vl-n. 

DIRECTIONS. 

DIVISOX  OF  TI.MK.  One  lliinl  of  tin-  aritliiiu-tic  time  alloliiH-iit 
of  tliis  yraile  .sliouM  l»e  devoted  to  oral  work.  One  llnnl  of  the  time 
should  be  devoted  to  couerete  problems. 

REVIEW.  Tal)les  of  linear,  dry  and  licjiiid  measure;  avoinlupois 
weight;  tiine,  and  U.  S.  money;  fractious;  aliijuot  jtarts  of  one-iunid- 
red.  aud  the  reading  atid  writiug-  of  deeimals  through  thousandths. 
Keej)  up  drill  on  the  fun  laneutal  oi)eratious  for  speel  and   aeeuracy. 

MAIN  TOPICS.  Denominate  numbers;  one  stej)  reiluetion.  aildi- 
tion,  subtraetiou,  multiplication,  and  division. 

Decimals;  notation  aud  numeration  to  ten-thousandths,  addition, 
subtraction,  multii)lication  and  division. 

DEXOMLXATE  NUMBERS.  The  work  should  be  made  objective. 
Show  that  reduction  asceudiug  aud  deceuding  is  similar  to  the  reiluct- 
ion  of  fractions  to  lower  aud  higher  terms  or  to  the  reduction  of 
whole  numbers.  The  reduction  of  2  bu.  1  pk.  to  quarts  is  similar  to 
changing  I,  I  to  l'2ths  in  order  that  they  may  be  combined.  "Long 
written  problems  in  addition,  subtraction,  multiplication  and  division 
of  denominate  numljers  should  be  omitted.  Problems  involving  re- 
duction through  more  than  three  denominations  are  seldom  used  in 
tlie  business  world".    How  to  Teach  Arithmetic:   Brown  an<l  Coffman. 

DECIMALS.  Adilition  and  subtraction  of  decimals  offer  no  diHi- 
culties  exce])t  those  previously  met  in  the  same  process  with  integers. 
In  multiplication,  teach  that  the  number  of  places  in  the  i»roduct  is 
the  sum  of  the  places  in  the  multii»lier  and  the  multiplicand,  and  that 
division  is  reverse  of  this.  However,  see  Forms  for  all  (trades,  for 
division  of  decimals.  The  clianging  of  fractions  into  decimals  should 
be  emphasized,  as  this  is  a  common  and  useful  jiractice. 


SUBJECT  MATTER. 

DEXOMIXATE  Xl'MBERS.     Ad<lition,     Subtraction.    .Multi|.lica- 
tion  and  Division. 

DECIMALS.       A(lditi(-ii.    Subtraction.    .M  iillijilicat  ion    and     Divi- 
sion. 

PERCEXTACE.      Construct  witli  the   children  ami    teach  the  fol- 
lowing table: 


Common  Fractions 

Decimal  Fractions 

Parts  of  $1.00 

Per  Cents 

50 

1 

100 
25 

.5 

>! 

.50 

50 

1 
4 

100 
331 

.25 

-S 

.25 

25 

1 
S 

100 
20 

.33i 

>! 

.331 

331 

I 

100 
16j 

.2 

X 

.20 

20 

1 
6 

100 
12.1 

.161 

8 

.16^ 

11)^ 

1 
if 

100 
10 

.121 

8 

.12i 

12.^ 

iV 

100 
75 

.1 

8 

.10 

10 

3 

4 

100 

.75 

¥» 

.75 

75 

OPTIONAL  WORK.    Work  for  higher  degree  of    accuracy. 
Solution  of  simple  three  step  problems. 


GENERAL  PROJECT. 

This  is  a  suggested  project  which  will  involve  much  of  the  sub- 
ject matter  of  the  grade.  Lest  it  should  not  provide  sufficient  drill 
to  fix  the  processes,  minor  projects  are  given  as  details  under  each 
topic  of  subject  matter.  The  latter  are  unrelated,  but  might  be  an 
outgrowth  of  a  larger  project  similar  to  the  one  here  given. 

What  profit  could  be  made  if  twenty  vacant  lots  in  the  vicinity  of  the 
school  house  were  cultivated  as  gardens  ? 

FRACTIONS,  DENOMINATE  NUMBERS,  DECIMALS.  There  are 
16  lots  in  1  city  block.  How  many  blocks  in  20  lots  ?  Find  answer 
to  3  decimal  places. 

If  there  are  2.75  acres  in  1  block,  how  many  acres  in  the  20  lots? 

If  .25  of  a  lot  makes  one  garden,  how  many  gardens  can  be  made 
on  the  twenty  lots? 

.Make  a  drawing  of  the  gardens  on  1  lot,  50  by  150  ft.,  using  a 
scale  of  1  inch  =^  12.5  ft.  and  leaving  walks  3  ft.  wide  around  each 
garden. 

If  fertilizer  is  used  at  the  rate  of  250  lbs.  per  lot,  how  much  will 
l>e  re<iuired  for  the  twenty  lots? 

What  will  it  cost  at  §30  per  ton? 

How  much  will  it  cost  to  get  it  from  the  depot  if  the  truck  driver 
charges  !?:i.50  a  load  and  carries  3  tons  per  load? 

If  labor  costs  55c  i)er  hour  and  an  ordinary  workman  spreads  at 
the  rate  of  2  tons  an  hour,  what  will    it    cost  to  i)ut  the  fertilizer  on  ? 

What  will  be  the  entire  cost  of  putting  on  the  fertilizer  ? 


i 


73 

How  lon<>'  will  it  take  a  man  to  jilow  tlu'  ciitii't'  area  it"  lie  |)l(>ws 
2.5  acres  ])er  day? 

What  will  l>e  the  cost  of  iilowiiitr  it"    he    ehartres  >!l.l'2  jier  hour":' 

Wliat  is  the  entire  cost  of  t"ertili/in<i"  and  iilowinu'  tin-  entire  area? 

What  is  the  cost  per  garden  ? 

What  is  the  cost  per  lot  ? 

If  you  should  decide  to  plant  o  lots  to  ])otatoes,  how  many  will 
he  required  if  each  lot  takes  8  bu.  2  pk.  18  (jt.  of  potatoes? 

What  will  he  the  cost  of  the  potatoes  at  -^1.15  per  bushel? 

Mow  many  hills  of  potatoes  can  be  ])laiiteil  on  I  lot.  7")  by  loO 
feet,  if  they  are  planted  in  rows  3.5  ft.  apart  ? 

What  will  be  your  yield  of  potatoes  if  your  lots  averajj^e  45  bu. 
8  pk.  ?  "  ' 

What  are  these  worth  at  x2.25  i)er  bushel? 

What  will  be  your  net  protit  after  de<lucting-  cost  of  fertilizinjc. 
plowing-  and  planting? 

If  you  should  decide  to  i)lant  3  lots  to  strawberries,  how  many 
plants  can  you  put  on  each  lot  if  they  are  placed  1.5  ft.  apart?  How 
many  can  be  put  on  three  lots? 

What  will  they  cost  at  15c  per  plant? 

How  many  lots  could  you  plant  to  strawberries  using  your  net 
protit  from  the  potatoes? 

A  strawberry  bed  will  yield  for  2  years.  If  the  plants  cost  !!<2.00 
an<l  the  fertilizer  §1.00,  how  much  is  made  from  the  bed  in  3  years 
(1st  year  it  does  not  produce  any  berries)  if  it  yiehls  (50  quarts  per 
year  at  an  average  price  of  10c  per  (juart  ? 

Find  the  cost  of  fencing  a  20  ft.  sq.  strawberry  jiatch  at  -s4.5() 
per  nl. 

From  2  (jts  of  pea  seed,  2  bushels  of  peas  were  obtained.  'J'he 
seed  cost  35c  per  quart  and  the  fertilizer  about  20c.  How  much  was 
gained  when  green  peas  were  selling  at  an  average  jirice  of  25c  per 
(piart  ? 

A  package  of  carrot  seed  costs  10c.  The  crop  was  al)out  3  pecks. 
What  was  saved  on  carrots  if  they  were  selling  at  an  average  ])rice  of 
5c  per  (juart  ? 

The  asparagus  bed  cost  -si. 50  for  the  roots  and  about  -si. 0(1  for 
ihe  fertilizer.  Each  year  about  <S  bunches  were  cut  from  it  at  an 
average  price  of  25c  per  bunch.  What  would  be  the  gain  in  5  years, 
allowing  25c  per  year  for  fertilizer?  The  tirst  year  it  was  jilanteil  m> 
asparagus  was  cut. 

A  dozen  cabbage  plants  cost  20c.  They  re(|uired  about  15c 
worth  of  fertilizer.  10  heads  were  obtained  that  were  worth  an 
average  price  of  lOc  each.    Find  the  gain. 

From  2  (luarts  of  onion  sets  that  cost  15c  jier  quart,  alioul  >^l.2.'> 
worth  of  onions  was  obtained.    Find  the  gain. 

The  corn  yielded  about  20  dozen  ears  at  an  average  price  of  20c 
per  dozen.  The  seed  cost  15c  ami  the  fertilizer  about  50c.  What 
amount  was  saved  on  corn? 

Find  the  total  amount  savetl  ? 


74 

lli.w  many  busliels  of  ouioiis  can  lie  l»(iu<:ht  for  ><'20.S0  at  $  .'20  a 
lialf  pei-k. 

Fiu«l  the  aiuouut  of  the  followiuir  Vtill: 

1  pk.  of  potatoes  at  81.60  per  l>u. 

2  pt.  cream  at  8.S0  per  <it. 

1  qt.  ci<ler  viuegar  at  8.60  per  yal. 
4  o/.  mustard  seed  at  8.40  jier  11). 

The  above  are  ouly  suggestive  minor  problems.  The  class 
should  decide  upou  what  they  wish  to  plant  in  the  various  gardens 
and  figure  accurately  the  cost  and  amount  which  they  might  reason- 
ably expect  from  an  average  yield. 


MINOR  PROJECTS. 

DENOMINATE  NUMBERS. 

To  find  how  many  years,  months,  and  days  old  I  am  to-tlay. 
To  find  the  difference  in  the  length  of  time  it  takes  two  steamers 
to  cross  the  ocean.  One  makes  the  trip  in  6  days.  20  hrs.  15  min.  and 
the  other  in  5  days,  15  hrs.  20  min. 

To  find  the  length  of  fence  needed  for  my  garden.  Estimate, 
then  verify. 

To  find  how  much  milk  is  used  by  the  family  for  one  week. 
To  find  the  amount  of  the  following  bill: 
1   pk.  of  potatoes  at  81.60  per  bu. 
I   pt.  cream  at  8. SO  per  qt. 
1   qt.  cider  vinegar  at  8.60  per  gal. 
4  oz.  mustard  seed  at  8.40  per  lb. 
To  find  the  number  of  tons  of  coal  in  the  following  deliveries: 

62,500  lbs.,       74,560  lbs.,       84,240  lbs.,       66,720  lbs. 
To  find  the  cost  of  a  city  lot  having  a  frontage  of  2S  ft.  8  in.  at 
832.50  per  running  foot. 

To  find  the  cost  of  fencing  my  yard  at  84.50  ]jer  rd. 

DECIMALS.  'i'o  count  by  6  tenths.  Begin  with  8  tenths  deci- 
mall.r  and  coimt  by  (5  tenths  until  told  to  stop.  Write  the  numbers 
while  counting. 

To  count  by  300  thousandths.  Begin  with  8  thousandths  and 
count  by  800  thousandths  until  tohl  to  stoj).  Write  numbers  while 
counting. 

To  find  the  distance  through  the  Panama  Canal.  A  ship  enter- 
ing from  the  iVtlantic  Ocean  travels  7.8  miles  an<l  is  then  raised  to  the 
high  part  of  the  canal,  Thru  this  i)art  the  shij)  goes  82.0(5  miles,  is 
lowered,  and  then  goes  10.6  miles  farther  to  Panama  City. 

To  find  the  difference  in  length  of  the  Suez  ami  Panama  canals 
if  the  Suez  canal  is  00  miles  long. 


To  tiinl  the  weifflit  of  tlie  packsack  yoii  take  on  your  cam  pint; 
trip.      Vol!  take  the  foHowiuy: 

5  pktjs.  of  choeohite  eacli  \vei«jliiiii>-  .'4'!  lli. 

6  cans  of  coudeused  milk,  each  wei'rliin},'-  .*2S  11). 
B  pkffs.  of  tea  each  weifrhiufr  .09  11>. 

4  pkgfs.  of  canned  meat,  each  weiffhiny  1.1 :{  lb. 
30  yds.  of  rope  weig-hinfj  .17  pounds  i)er  yard. 
4  cans  of  oil,  weighing  1.07   i>ound8  i)er  can. 
To  find  the  number  of  bushels  of  onions  that  can  be  bought.     At 
>^.'20  a  half  peck,  how  many  bushels  of  onions  can  be  bought  for  -s'iO.SO':' 
To  find  the  length  of  fence  around  my  garden   in   feet   and   ro<ls. 
The  length  of  fence  around  my  garden  is  .25  of  a  mile. 

To  find  the  cost  of  887  lb.  of  sugar  at  >!ll.45  per  C'wt. 
To  find  the  average  rate  of  speed  per  hour  of  a  warship.     A  war- 
shi])  makes  the  following  record  in  four  hours:  in  the  first  hour,  19.5 
mi.,  in  the   second   hour.   21.75   mi.,   in   the  third  hour.   22.2   mi.,   in 
the  fourth  hour,  22.9  mi. 

Standards  of  Attainment. 

The  child  should  have  the  habit  of  estimating  results  well  fixed. 

He  should  have  acquired  reasonable  speed  and  accuracy  in  the 
fundamental  operations  of  fractions  and  decimals. 

He  should  know  thoroughly  all  denominate  number  tallies  taught 
thus  far. 

He  should  be  able  to  read  and  write  decimals  rajtiilly  and  with 
assurance. 

Objective  Standards. 

See  Grade  tJA. 

Bibliography. 

MKTHOD. 

J>rown    and   Coffman:  How    to    Teach    Arithmetic; 

Denominate  Numbers,  Chap.  XII; 

Decimals,  Chap.  XIV. 
Klapjier:  The  Teaching  of  .\i-ithmetic; 

Decimals,  ('ha]).  IV.,  p.  7S.  ]»i).  242-258. 
Wilson  and  Wilson:  Motivation  of  School  Work; 

Chap.  IX,  Motivation  of  Arithmeiic. 

SIPI'LEMENTARV  BOOKS. 

Iloyt  and  Peet:   Everyday  Aiithmctic-.  Txink  II: 

Denominate  Xumliers  Part  III.  Chap.  \'II: 

Decimals.  Part  II.  Chap.  \'I: 

Part  IV.  Chaj..  III. 
Stone-Millis:   Intermediate  Arithmetic-; 

Decimals,  pp.  1(58-1X5. 
Wentworth-Smith:   Essentials  of  Aritlimetic.  Intt-rme  liate  ISouk; 

Denominate  Xumbers  Chap.  Ill: 

Decimal  Fractions.  Chaii.  I\'. 


GKADE  VI- A. 

DIRECTIONS. 

L)l\"lSION  OP'  TIME.  Due-thinl  uf  the  arithmetic  time  aUotmeut 
of  this  jjrade  shouhl  be  devoted  to  oral  work. 

( )iie-half  of  the  time  should  be  given  to  concrete  problems. 

REXIKW.  The  fundamental  operations  in  fractions  and  decimals; 
all  denominate  number  tables;  one  step  reductions;  simple  percentage 
tables.  Many  oral  concrete  problems  should  be  used  in  the  review. 
MAIN  TOPICS. 

SQUARE  MEASURE  AND  CUBIC  MEASURE. 

Square  measure:  Develop  and  teach  the  table  of  square  measure. 
The  idea  of  squares  should  be  developed  by  drawings,  by  foMiug 
paper,  and  with  mathematical  blocks.  The  child  should  have  a  clear 
mental  picture  of  a  square  inch,  a  square  foot,  a  square  yard,  a  square 
rod  and  an  acre.  Accurate  diagrams  drawn  to  a  scale  should  be 
made.  Teach  terms  rectangle,  quadilateral  triangle,  base,  altitude, 
perimeter  and  area.  The  area  of  a  triangle  equals  one-half  the  area 
of  a  rectangle  having  the  same  base  and  altitude. 

Cubic  measure:  Develop  and  teach  the  table  of  cubic  measure. 
Develop  finding  the  volume  of  rectangular  solids  with  inch  cubes  and 
other  mathematical  blocks.  Make  practical  applications  of  volume; 
hauling  coal,  selling  wood,  capacity  of  bins.  Solution  of  prol)lems 
shoulil  be  deferred  until  all  steps  liave  been  indicated.  The  children 
should  be  taught  to  estimate  results  and  be  able  to  tell  when  results 
are  impossiltle.  Make  clear  concepts  in  order  that  insight  into  fu- 
ture work  will  be  accurate.  The  clear  concept  is  the  important  thing 
in  this  elementary  beginning  of  mensuration. 

Hoard  measure:  Pieces  of  lumber  should  be  brought  to  class 
from  the  shops,  measured,  and  the  number  of  board  feet  computed. 
The  children  should  visit  a  lumber  yard  or  a  house  and  note  the 
kinds  of  lumber  used.  Use  the  process  that  the  lumber  dealer  uses 
in  calculating  materials  required  in  building  a  house. 

DECIMALS.  Simple  three  step  problems  may  now  l)e  intro- 
duced. 

PERCENTAGE.  Changing  decimals  to  i)erceuts;  a  continuation 
of  tlie  jtercentage  tables  taught  in  VI-B. 


SUBJECT  MATTER. 

S(|uare  Measure. 
Cubic  Measure. 
IJoard  Measure. 

DECIMALS.      Notation  and  numeration  to  six   places  should   be 
developed  and  used. 


PERCENTAGE.      Develo])  and  teacli  the  t'()llowiii<r  as  a  contiima- 
tiou  of  the  tables  begun  in  the  N'l-ll. 

Common  Fractions  Decimal  Fractions  Part  of  ^1,00  Percent 


100 

1 

66:1 
100 

.661 

S.66g 

66^ 

0 

40 
lOU 

.40 

8.40 

40 

3 
5 

GO 
100 

.60 

>^.60 

60 

4 

5 

SO 
100 

.80* 

8.80 

80 

?, 

100 

.m 

8.831 

m 

3 

ft 

100 

.87i 

8.87^ 

87i 

5 

100 

.62i 

8.62i 

62i 

8 

87i 

.87^ 

8.87i 

87  i 

100 

OPTIONAL  WORK.  Solution  of  more  difHcult  problems.  Drill 
tor  more  speed  and  accuracy  in  fractions  and  decimals. 

GENERAL  PROJECT. 

This  is  a  sujjfoesteil  project  which  will  involve  much  of  the  sub- 
ject matter  of  the  jjratle.  Lest  it  should  not  provide  sufficient  drill 
to  fix  the  processes,  minor  projects  are  given.  The  latter  are  un- 
related, but  might  be  an  outgrowth  of  a  larger  project  similar  to 
the  one  here  given. 

TO  FIND  THE  COST  OF  IMPROVING  A  NEARBY  VACANT  BLOCK. 

It  will  first  be  necessary  for  the  class  to  decide  what  sort  of  im- 
provement they  prefer  to  make.  This  will  depend,  of  course,  some- 
what upon  the  character  and  location  of  the  block.  There  may  be  a 
block  which  is  wooded  and  could  easily  be  made  a  park,  or  it  may  be 
ojten  and  suitable  for  skating  rink.  Whatever  the  class  may  decide 
upon  may  be  made  tlie  basis  of  the  ])roject  and  the  teacher  may  di- 
rect the  develoi)ment  of  minor  i)rojects  and  problems  to  suit  (he 
mathematical  reijuirements  of  the  grade. 

In  the  vicinity  of  one  of  our  schools  is  a  bloek  which  has  already 
been  set  aside  by  the  city  as  a  park.  It  already  has  a  skating  rink 
and  some  benches  and  tables  and  two  or  three  buildings.  It  would 
be  an  excellent  community  project,  however,  to  imjtrove  this  further. 
In  case  a  class  in  that  school  should  decide  to  take  (he  further  im- 
provement of  that  park  as  a  project,  the  following  minor  projects 
might  naturally  come  out  of  it. 


78 

SQUARE  MEASl'RE,  BOARD  MEASURE.  FRACTION'S,  DECIMALS. 
Make  a  Vml  for  the    contract    to    put    a    cenieut   sidewalk  arouud  the 
])lock.     Let  jHipils  do  this  iudepeudeutly  at  tirst,  theu    work  it  out  as 
a  class.   The  habit  of  approxirnatiuj?  auswers  may  l)e  greatly  streujjth- 
eiieil  throuyhout  this  project. 

Find  the  cost  of  fencing.  Decide  on  kind  of  fence  and  figure 
lumber  and  labor  cost;  kind  of  fence  and  figure  painting  fence,  V)uild- 
iugs,  l)enches  and  tables,  if  it  requires  1  gallon  to  give  250  sq.  ft.  one 
coat,  and  paint  costs  §2.50  per  gal.  Which  would  cost  less,  the  fence 
or  a  hedge  if  the  hedge  plants  cost  *6  per  100  and  are  set  10  inches 
apart  ? 

Find  the  amount  of  grass  seed  necessary  to  seed  it,  deducting  the 
area  of  the  rink,  if  1  qt.  of  seed  is  sufficient  for  300  sq.  ft.  Find  the 
cost  of  seeding.  Let  pupils  find  out  the  cost  of  seed,  labor,  the 
length  of  time  which  it  would  be  likely  to  take. 

Find  the  amount  of  lumber  necessary  for  making  more  tables 
and  chairs.  Decide  on  the  kind  of  lumber  to  be  used  and  let  pupils 
find  out  the  prices. 

It  may  be  decided  to  put  in  posts  for  electric  lights.  If  they  are 
9  ft.  high  and  4  inches  S(iure,  how  much  lumber  will  it  lake  for  1 
dozen  jxjsts? 

What  will  be  the  cost  of  giving  them  2  coats  of  paint? 

How  much  earth  will  be  dug  out  for  setting  them  if  they  are  set 
14  in  dies  deep  '? 

What  will  be  the  cost  of  putting  arc  lights  on  these  posts  at 
^4.70  per  light?  If  each  light  burns  500  watts  per  hour,  how  many 
watts  will  all  consume  in  one  evening? 

What  will  the  cost  of  lighting  per  evening  if  electricity  costs  10c 
per  1000  watts  ?     What  will  be  the  yearly  cost? 

Figure  the  cost  of  making  3  or  4  flower  beds.  How  many  canna 
plants  will  be  necessary  for  a  rectangular  bed  6  ft.  x  30  ft.  if  they  are 
set  2  ft.  apart  in  rows?  What  would  they  cost  at  §8  per  100  plants? 
What  would  they  cost  at  §1.25  per  dozen  plants? 

Find  the  number  of  feet  of  curbing  around  the  entire  block  if  it 
is  400  ft.  X  300  ft.  ?  What  would  be  the  expense  of  curbing  at  65c 
per  running  foot  ?  Suppose  this  expense  were  shared  by  the  property 
owners  across  the  street  acconling  to  the  number  of  feet  frontage 
they  secured.  Find  the  share  of  each  of  the  following:  A  owned  198 
feet,  K  5S  ft.,  C  6<S  ft.,  D  150  ft.,  E  75  ft.  and  F  72  ft.  6  inches. 

Estimate  the  amount  of  coal  necessary  for  heating  the  warming 
house  during  the  months  of  December,  January  and  February. 

How  large  a  bin  must  be  built  to  hold  sufficient  coal  for  the  three 
months  if  1  ton  contains  about  36  cu.  ft.  ?  How  much  would  be 
saved  by  filling  the  bin  in  April  when  coal  is  50c  cheaper  than  in 
winter? 

If  the  care  taker  oversees  the  skating  rink  from  7:15  until  9:30 
six  nights  in  the  week,  what  is  his  entire  time  for  the  week.  What  is 
his  time  for  the  three  months  ?  How  much  money  will  he  draw  if 
he  gets  45c  per  hour? 

If  a  party  of  20  people  engage  the  rink  for  an  evening,  what 
would  be  each  person's  share  of  the  caretaker's  wage  ?     What   would 


.79 

l)e  a  fair  amount  for  the  city  to  charg-e  for  the   heatiiitr  and  liirhtiiit,'":' 
A  part  of  this  hlock  is  low  and  needs  to  he  raised  on  theaverHfje 
of  6  ft.     At  50c  per  cu.  yd.,  what  will  it  cost  to  raise  this  area  which 
is  40  ft.  by  20  ft.  ? 

MINOR  PROJECTS. 

SQUARE  MEASURE. 

To  find  the  cost  of  varuishingr  your  desk  at  so  much  per  scpiare 
yard. 

To  find  the  numher  of  square  feet  of  blackboard  in  the  room, 

To  find  how  much  it  would  cost  to  plaster  the  walls  and  ceilinfj 
of  a  room. 

To  find  the  area  of  a  city  lot  if  it  has  a  frontage  of  50  feet  and  a 
depth  of  100  feet. 

To  find  the  length  of  an  acre  of  land  if  it  is  10  rds.  wide. 

To  find  the  cost  of  fencing  the  above  field  with  wire  fencing  at 
10c  a  foot. 

To  find  the  number  of  square  yards  of  material  it  would  take  to 
make  a  triangular  pennant  whose  base  is  one  yard  and  whose  altitude 
is  seven  feet. 

To  find  the  cost  of  sodding  a  lawn  35  feet  by  45  feet,  if  the  soils 
are  1<S  in.  by  80  in.  and  cost  §.08  a  piece. 

CUBIC  MEASURE. 

To  find  number  of  cubic  feet  of  air  in  the  room. 

To  find  the  number  of  tons  of  hard  coal  that  can  be  put  into  a 
bin  7  ft.  by  5  ft.  by  3  ft. 

To  find  the  number  of  cords  of  wood  in  a  pile. 

To  find  the  number  of  bricks  that  were  needed  to  build  one  side 
of  the  school. 

To  find  the  number  of  loads  of  earth  nee<led  to  raise  the  school 
yard  a  foot. 

To  find  the  number  of  gallons  of  oil  in  a  tank  4  ft.  wide,  0  ft.  3 
in.  long  if  the  oil  is  6  ft.  deep. 

To  find  the  number  of  barrels  of  oil  in  the  above  tank. 

To  be  able  to  read  and  understand  the  rainfall  table. 

To  make  a  graph  showing  the  rainfall  in  Duluth  by  seasons,  dur- 
ing the  past  year. 

To  find  the  cost  of  putting  cement  sidewalks  in  front  of  a  lot. 
The  lot  is  50  by  140  feet  and  the  walk  is  to  be  ii  ft  wide.  The  best 
walk  costs  §1.65  a  square  yard;  a  good  ([uality  costs  «!. 50,  and  «-enieut 
blocks  can  be  laid  for  §1.25  a  stjuare  yard.   Which  kind  shall  be  put  in? 

BOARD  MEASURE. 

To  find  the  cost  of  lumber  needed  for  a  fence,  a  sled,  a  window 
box. 

To  find  the  cost  of  40  pieces  of  lumber,  2"'x4''  and  14  ft.  lout:  at 
§35  per  M. 


To  fiu<l  the  cost  of  timber  for  the  fouudation  of  your  house  if  you 
uee<l  10  timbers  8"xl'2"  aud  16  ft.  loug  at  x4U.OO  per  M. 
To  tiu<l  the  cost  of  a  footstool  for  mother: 
4pc8.  Oakl.rxl|"x8"     ) 

2    "        "     2l"xl"xl0"     )      Oak  costs  10c  a  board  foot. 
•2    "        "     '2V'xr\\7"        ) 

1    "      Piue  7"xlO'\\r*  — costiug  7c  a  Bd.  ft. 

Hi  screws  at  lOc  i)er  doz. 
lOc  for  staiu. 


Standards  of  Attainment. 

The  child  should  have  the  power  to  see  relationships. 

lie  should  be  able  to  express  accejitably  the  solutions  of  his 
l»roblems  orally  and  in  writing:. 

The  child  should  have  the  assurance  of  whether  he  is  rifjht  or 
wrong  because  of  well  controlled  means  of  checking  his  work. 

He  should  have  a  well-defined  concei)tion  of  square  and  cubic 
measures. 

WOODY  TEST:  SERIES  A 

Addition 29.7 

Subtracticm 24.9 

Multiplication 26.1 

Division 23.7 

COURTIS  TEST  Series  B-Speed 

Addition 10 

Subtraction 11 

Multiplication 9 

Division 8 

COURTIS  TEST  Series  A-Accuraey 

Addition 73 

Subtraction 85 

M  ultiplication 78 

Division 87 

STONE  REASONING  TESTS  Actual  Medians  Obtained 

Butte,  1914 ^_ 3.9 

Bridgeport,  Conn.,  1913 5.2 

Salt  Lake  City,  1915 6.4 

Nassau  Co.  N.  Y.,  1918 4.5 

Lead,  8.  Dak.,  1916 6.1 

Tentative  Standards  (1916). 

That  80  i)er  cent  or  more  of  the  Sixth  grade  pupils  reach  or  ex- 
ceed a  score  of  6.5  with  at  least  80  i)er  cent  accuracy. 


81 
Bibliography. 

MICTllOl). 

I)i-()\vii  and  C'offinaii:    How  to  Teat-h   Arithmetic; 

C'liaj).  XXI,  Mensuration. 
Klai)i)er:  The  Teacliinii  of  Aritlinietie; 

pp.  309-820. 
McMurray:   ISpecial  .Method  in  Aritlinietic; 

Chap.  IV,  Method  in  Intermediate  Classes. 
Wilsou  aud  Wilsoja:  Tlie  Motivation  of  School  Work; 

Chap.  IX,  Motivation  of  Aritlinietic. 

SUPPLEMENTARY  BOOKS. 

Hoyt  and  Feet:   Everyday  Arithmetic,  IJook  Two; 

Part  Four,  Chap.  VII,  Measurements; 
Chap.  VI,  Perceutag-e; 
Everyday  Arithmetic,  Book  Three; 

Part  Five,  Chaj).  VIT,  Measurements; 
Chaj).  Ill,  Percentage. 
Stone-Millis:  Intermediate  Arithmetic; 

Chap.  VIII,  Mensuration; 

Chap.  VII,  Percentage 
Thorudike:  Arithmetic,  Book  Two; 

Chap.  IV,  Measurements; 

Chap.  Ill  and  V,  Percentage; 

Book  Three;  pp.  108-110,  Board  Feet; 

Chap.  V,  Measurements. 
Wentworth-Smith:  Essentials  of  Arithmetic; 

pp.  188-209,  Measuring-; 

Chap.  VII,  Percentage. 


82 
GRADE  VIM}. 

DIRECTIONS. 

I)I\  ISION  OF  TIME.  One-fourtli  of  tl)e  time  of  tliis  tirade  should 
be  devoted  to  oral  work.  Seveii-eio^hths  of  the  time  should  he  de- 
v<)te<l  to  eoucrete  problems, 

RKXIEW.  Decimals,  fractious,  aliquot  parts  of  a  dollar  with 
fractional  equivaleuts.  chantriutj  fractious  to  decimals  and  decimals  to 
fractious,  aud  writiug  decimals  as  per  ceuts. 

Drill  for  accuracy  aud  speed  in  uotatiou,  uuiueratiou,  addition, 
subtraction,  multiplication  aud  division. 

Denominate  numbers — Oral  problems  for  accuracy  and  speed  in 
solving  problems  related  to  the  daily  transactions  of  business  life. 

LE.ADING  TOPIC.     Percentage. 

PERCENTAGE.  One  of  the  most  frequent  sources  of  error  in  per- 
centage is  due  to  the  failure  of  the  pupil  to  keep  clearly  in  mind 
per  cent  of  what.  In  all  early  work  in  percentage  the  teacher  should 
re(|uire  the  pupil  to  name  the  quantity  on  which  the  per  cent  is  based. 
Tliere  are  no  new  mathematical  principles  involved  in  percentage.  It 
is  only  the  terminology  that  makes  it  hard  for  the  child.  This  subject 
should  be  taught  as  a  continuation  of  fractions.  Commercial  dis- 
count, profit  and  loss,  aud  commission  should  be  introduced  as  appli- 
cations of  percentage.  The  only  difference  between  profit  and  loss 
and  commercial  discount  is,  that  one  is  based  on  the  cost,  the  other  on 
the  list  or  marked  price.  Teaching  terms  other  than  'rate'  is  confusing 
and  should  not  be  done.  In  connection  with  percentage,  teach  the 
algel)raic  equation,  using  the  term  "X",  in  place  of  the  unknown 
(piantity.  After  the  equation  has  been  introduced  it  should  be  used 
whenever  it  will  aid  in  the  solution  of  arithmetical  problems.  All  re- 
sults should  lie  checked  by  the  child.  This  gives  him  confidence  in 
his  own  al)ility  and  develops  habits  of  accuracy.  Budgets  of  house- 
hold expenditures  and  savings  should  be  made  in  connection  with 
I»ercentage. 

SUBJECT  MATTER. 

PERCENTAGE.  As  pertainmg  to  commercial  discount,  profit 
aud  loss,  and  commission.  The  two  important  processes  in  the  sub- 
ject of  percentage  are:  1.  Finding  some  per  cent  of  a  given  num- 
ber. 2.  Findiug  what  per  cent  one  number  is  of  another.  All  else  is 
relatively  unimportant  aud  on  these  two  the  emphasis  should  accord- 
ingly be  place<l. 

OPTIONAL  WORK.  Solution  of  more  difticult  problems.  Some 
very  simple  work  in  grai)hing. 

GENERAL  PROJECT. 

These  are  suggested  projects  which  will  involve  much  of  the  sub- 
ject matter  of  the  grade.  Lest  they  should  not  provi<le  sufficient  drill 
to  fix  the  processes,  minor  projects  are  given.     The  latter  are  unrelated 


S3 

but  might  be  au  uuttirowlli  of  larger  projects  similar  to  tlie  ones  here 
giveu. 

TO  ESTIMATE  THE  COST  TO  THE  CITY  OF  TAUDINESS  AND   ABSENCE. 

PERCENTAGE.  The  cost  of  education  in  the  U.S.  for  191G  was 
§914,<S04,171  of  which  amount  ^S.'jS, 891, 3(54  was  spent  for  public  ele- 
mentary schools.     What  percent  was  that  of  the  entire  amount':' 

The  cost  for  public  high  schools  was  !?!<S2,3"2.'),()S9.  How  much 
greater  percent  was  spent  for  elementary  schools  than  for  high  scho(jls? 

The  total  number  of  elementary  and  high  school  children  in  the 
U.  S.  in  191(5  was  estimated  to  be  19,704,209.  What  was  the  cost 
per  cai)ita":' 

If  there  were  that  year  14, 90S  i)ui)ils  in  the  elementary  ami  high 
schools  of  Duluth,  what  percent  of  this  entire  amount  was  spent  in 
Duluth? 

In  1917  the  cost  of  text  books  in  your  school  was  S2S6.9(i.  What 
percent  of  the  entire  amount  was  expended  for  each  pujiil  if  there 
were  53S  pupils? 

The  State  Api)ortionmeut  was  §79,620. SO.  What  percent  of  this 
did  each  pupil  draw  for  attending  school? 

In  1917  the  enrollment  in  Duluth  was  15,725.  The  cost  of  main- 
tenance was  as  follows: 

Wages  of  janitors  and  helpers §44,970.29 

Fuel 29,S30.4(3 

Water 3,57 1 .45 

Light  and  power 2,957.3S 

Janitors'  supplies 3,245.42 

Services  other  than  personal 1,107.22 

Kepairs  and  Upkeep 1S.501.SS 

Instruction 475,9t)0.93 

Find  total  cost  per  capita. 

Find  cost  of  instruction  per  capita. 

What  percent  of  entire  amount  was  si)ent  for  instruction? 

What  is  the  money  loss  to  the  district  if  1  puinl  is  absent  12  days 
during  a  semester? 

Reckon  the  amount  of  money  you  have  lost  to  the  district  by 
absence  and  tardiness. 

What  percent  is  it  of  the  amount  the  district  expended  for  you? 

What  is  the  percent  loss  for  the  entire  room? 

What  does  it  cost  the  district  if  you  fail  your  grade  and  it  is 
necessary  for  you  to  repeat? 

Find  out  how  great  a  per  cent  of  the  entire  1917  cost  was  lost  to 
the  district  in  your  school  if  there  was  a  jiercentage  of  failure  of  3 
per  cent? 

2.  To  tind  the  jtrotits  a  boy  would  make  out  of  a  garden  iilot  if  his 
father  gave  him  a  certain  per  cent  of  the  crop  for  heli)ing  to  culti- 
vate it.  If  your  father  gave  you  15  per  cent  of  his  crop  of  potatoes 
for  helping  to  cultivate  it,  how  many  bushels  would  you  receive,  if 
he  raised  40  bushels? 

If  your  father  sold  the  jxitatoes  at  §2  a  bushel  what  was  your 
share  of  the  money? 


84 

What  was  your  share  of  the  i)rolits  if  5  I'li-  (>f  potatoes  at  -^1.00 
lier  liu.  had  been  useil  iu  ithiutiiitjy 

If  from  the  frardeu  you  used  euougli  jjroduce  to  reduce  your 
weekly  exjjeuse  account  from  §14.50  to  811.25,  what  per  ceut  was 
saveil':' 

I  low  much  mouey  did  you  contribute  to  the  family  table  if  15 
l»er  cent  t»fthe  food  used  belonfjed  to  you? 

If  you  sold  yarden  jiroduce  for  your  father  on  a  commission  of 
80  per  ceut,  what  would  you    earn  in    1  day  from  the  following  sales: 

Radishes,  10  bunches at    5c 

Onions,  20  bunches at    7c 

Peas,  6  lbs at  15c 

Beans.  10  lbs at  15c 

8.  To  tind  what  per  ceut  of  the  games  played  by  the  Junior  High 
School  base  ball  leag'ue  our  team  won.  The  Junior  Ilig'h  School 
base  ball  league  played  20  games  in  all.  Our  team  won  18  games. 
What  per  ceut  of  the  20  games  did  they  win?  Use  as  a  cheek  to 
the  above  problem.  Find  how  many  g-ames  were  played  by  the 
leag'ue.  Our  baseball  team  won  18  games  which  was  90  per  ceut  of 
the  whole  uuml)er  played.     How  many  games  were  played? 

To  rearrange  the  following  tables  according  to  the  excellency  of 
the  teams,  or,  To  tiud  how  our  team  stands  in  reg^ard  to  the  other 
teams. 

WEST  END  GRADE  SCHOOL 

WON  LOST 

Lincoln _7 2 

Lincoln  Cubs 0 8 

Bryants 8 2 

Moil  roe 3 6 

EAST  END    GRADE  SCHOOL 

WON  LOST 

Washburn 4 5 

Eudion 6 8 

t'ran  klin 5 4 

•Jefferson 3 6 

To  tiud  ])ei  ceut  of  games  won. 

To  grai)h  number  of  games  won  and  lost  by  schools. 

MINOR  PROJECTS. 

T(j  find  how  much  Current  Events  will  cost  a  class  of  forty  if  a 
discount  is  allowed  for  thirty  or  more. 

To  tind  liow  much  profit  has  been  made  from  the  garden. 

To  tiud  li(jw  much  is  made  or  lost  each  day  froTu  the  sale  of  news- 
l)ai)erH.    Find  the  jter  cent  of  gain  or  loss. 

'I'o  find  what  your  father  gained  when  he  sold  your  house  or 
farm. 

'i'o  Hud  the  jicr  cent  of  your  class  wiio  are  buying  thrift  stamps. 


So 

To  fiud  the  cost  of  a  bicycle  listed  at  a  certain  piirc,  uii  w  liicli 
two  successive  discounts  are  allowed. 

To  find  the  value  of  a  book  if  it  decreases  in  value  "i')  per  cent 
each  year  and  has  been  use<l  three  years. 

To  find  per  cent  of  discount  on  goods  daniage<l   by  tire  oi-  water. 

To  lind  which  you  would  rather  have,  if  you  were  a  salesman,  a 
stated  salary  or  a  wage  plus  commission  on   your  sales. 

To  investigate  fees  for  collecting  debts.  I  employeil  a  lawytT  to 
collect  a  debt  of  »i!3,(iOO.()0.  He  succee<led  in  getting  SU  per  cent  of 
it,  on  which  he  charged  a  commission  of  5  per  cent.  How  much  did 
I  receive  V 

To  find  how  much  is  saved  by  buying  in  large  quantities.  How 
much  does  your  mother  save  by  buying  canned  goods  at  «1.15  per 
dozen,  instead  of  one  at  a  time  for  l*ic. 

To  find  how  much  a  family  will  have  for  special  vacation  ex- 
penses. A  family  earn  §1,420.00  this  year.  If  they  spend  1(3  iier 
cent  for  car  fare  and  rent,  32  per  cent  for  food,  18  per  cent  for  fuel 
and  other  running  expenses,  10  per  cent  for  books,  music,  church, 
savings  and  insurance  10  per  cent,  how  much  will  they  have  for 
special  vacation  expenses?   (Problems  like  this  may  be  graphed.) 


Standard  of  Attainment. 

The  child  should  have  the  mastery  of  the  two  principles  of  jier- 
centage  as  applied  to  commercial  discount,  i)rotit  and  loss,  ami  com- 
mission. 

The  habit  of  checking  his  own  results  should  be  firmly  estab- 
lished. 

The  child  should  have  a  good  working  knowledge  of  all  terms 
used  thus  far  and  shouhl  be  able  to  express  himself  accurately. 

OBJECTIVE.      See  dirade  7-A. 


Bibliography. 

TEACHER'S  READING.    Hrown    and    (otfman:      liow    I-    Te.i.li 
Arithmetic. 
Chap.  XV.  Percentage. 

Klapper:  The  Teaching  of  Arithmetic. 

P]).  2o4-2GS.   Percentage. 

Pp.  287-290.  The  Equation. 
Wilson  and  Wilson:  Motivation  of  School  Work. 

Chap.  TX.   ^lotivatioii  of  Arithmetic. 


86 
SITPI.KMENTARV  BOOKS. 

Iloyt  and  Peet:    Kveryilay  Aritlimetic.     l>ouk  III. 

I'nrt  Five.   CMuip.  IX.  Town    and    City    IniprDveineut. 

C'liait.  VI,  J>aiikiii«i. 

Part  Six,     C'liaj).  XI.  The  Equation  ot"  Percentage. 
Stone  and  Millis:  A<]vanced  Aritlinietie. 

Chap.  IV.  and  VT.  Percentage. 
Thorndike:  Arithmetic.  Book  III.,  Chap.   II    and   III.    .Vjjplica- 

tions  of  Percentage. 
Wentworth-Smith:   Essentials  of  Arithmetic. 

Intermediate  Book.  Chai».  VII.  Applications  of  Percentage. 

Advanced  Book:  Chai).III.  Percentage  and  Its  Applications. 

Chap.  V.      Banks  and  Banking. 
Wentworth.  Smith  and  Brown:  Junior  High  School  ^Mathematics. 

Part  1. 


87 
(tKADE  VII-A. 


DIRECTIONS. 


DIVISION  OF  TIME.  One-fourth  of  the  aritliiiietie  time  allot- 
meut  of  tliis  tirade  should  l»e  devoted  to  oral  work.  Seveii-eijrhtlis  of 
the  time  should  l)e  devoted  to  concrete  problems. 

REVTPLW.     The  same  as  for  VII-]>,  usiiiy  more  ditticult  woik. 
Review  the  principles  of  i)ercentai>e  and   its  ajjplications. 
Use  equations  wherever  they  will  aiil  in  the  solution  of  the   pro- 
blems. 

LEADING  TOPICS.  Percentage.  Taxes,  duties  or  custi^ms.  in- 
surance, simple  and  compound  interest  should  be  introduced  as  further 
application  of  percentage.  The  informational  value  of  taxes,  duties  or 
customs,  and  insurance  should  be  emphasize*!  rather  than  the  math- 
ematical content. 

The  features  of  taxes  shoulil  be  presented  from  the  stand jioint  of 
civics  rather  than  that  of  arithmetic.  The  problems  should  be  based 
on  local  couditi(ms.  Make  the  work  simple.  The  pupils  should  be 
familar  with  the  terms;  valuation,  assessment,  assessors,  delinquent 
taxes,  levy,  rate  of  taxation,  and  board  of  equalization. 

The  subject  of  national  duties  and  revenues  is  closely  related  to 
that  of  taxes  and  may  be  presented  in  a  similar  manner.  The  math- 
ematical problems  should  be  brief. 

Only  the  common  types  of  property  and  i)ersonal  insurance 
should  be  considered.  The  terms,  ])remium,  policy,  endowment,  mat- 
urity, face  of  policy,  and  adjuster  should  be  taught.  The  forms  for 
l)olicies  should  be  obtained  in  order  that  the  work  may  be  ma  le  real 
and  practical.  Children  shouhl  be  shown  that  insurance  is  a  matter 
of  i)rotection  rather  than  of  money  making  for  the  policy  holder. 

"Pupils  sometimes  fail  to  master  the  subject  of  interest,  but  the 
failure  maj'  usually  be  traced  to  the  lack  of  an  accurate  understand- 
ing: of  the  terms  used  and  of  an  acquaintance  with  business  jiroceeil- 
ure  rather  than  to  any  mathematical  difficulties  involved.  There  is 
no  jjroblem  in  simple  interest,  the  solution  of  which  requires  a  degree 
of  mathematical  knowletlge  not  in  the  i)ossession  of  tlie  jmitil  who  is 
Ijrepared  to  begin  a  formal  study  of  the  suV)ject.  Simjile  interest  is 
an  easy  application  of  percentage  with  the  time  element  as  an  im- 
l>ortant  factor. 

"Numerous  definitions  have  been  suggested  for  the  term  interest. 
The  statement  that  "interest  is  money  rent"  is  i)robably  as  good  as 
any  that  have  been  proposed. 

"The  pupils  shouhl  understand  how  men.  when  leniling  money, 
require  a  certain  i»ayment  for  the  use  of  the  money.  The  teacher 
should  make  clear  to  the  pui)ils  how  a  man  can  aflord  to  borrow  a 
given  sum  for  a  year,  be  security  for  the  amount  liorrowed.  and  at 
the  end  of  year  pay  back  to  the  lender  not  only  the  amount  originally 
borrowed,  but  an  additional  amount,  which  is  called  interest.  Tin- 
sum  ui)on  which  the  interest  is  l>ased  is  called  the  i)rincip:il.  t..  dist- 
tinguish  it  from  interest. 


88 

"The  impilfi  slumlil  consider  the  various  factors  wliich  (leterniiue 
interest  rates.  Tlie  rehition  of  interest  rates  to  the  hnv  of  supi)ly  and 
demand  shoiihl  l)e  pointed  out.  Especial  attention  sliouhlbe  directed 
to  the  security  of  the  h)au  as  a  factor  in  deterniinintj  interest  rates. 
The  I'niteil  States  can  l)orrow  hirfje  sums  of  money  at  a  low  rate  of 
interest.  The  teacher  shouhl  impress  upon  the  pupils  the  fact  that 
very  hiyh  rates  of  interest  are  often  synonymous  with  poor  security. 
It  would  not  be  correct  to  say  that  high  rate  of  interest  are  always 
directly  associated  with  poor  security,  for  where  profits  upon  capital 
are  large,  the  rates  of  interest  are  high  as  a  result  of  the  law  of  supply 
and  demand.  However,  the  teacher  should  caution  the  pui)il  to  investi- 
gate with  more  than  usual  care  the  security  of  any  loan  when  very 
high  returns  are  promised.  In  this  connection  some  consideration 
shouhl  he  given  to  loan  sharks. 

"A  third  factor  that  determines  interestrates  is  the  time  for  which 
the  loau  is  made.  The  rate  is  usually  lower  upon  a  loan  for  a  long 
period  than  for  a  short  period.  The  amount  of  the  loan  is  also  a 
factor  in  determining  interest  rates.  The  rate  for  a  small  loan  is 
often  higher  than  for  a  large  one."  From  '"How  to  Teach  Arith- 
metic," Brown  and  Coffmau. 

Compound  interest  should  be  taught  only  in  connection  with 
savings  accounts  and  War  Saving  Stamps. 

Solve  many  interest  problems.  Examine  interest  tables  and 
solve  a  few  problems  to  show  the  use  of  them. 

SUBJECT  MATTER. 

PERCENTAGE.  As  applied  to  taxes,  duties,  or  customs,  insurance, 
simple  and  compound   interest. 

This  is  a  suggested  project  which  will  involve  much  of  the  sub- 
ject matter  of  the  grade.  Lest  it  should  not  provide  sutticient  drill  to 
fix  the  processes,  minor  projects  are  given.  The  latter  are  unrelated 
but  might  l)e  an  outgrowth  of  a  larger  project  similar  to  the  one  here 
given. 

GENERAL  PROJECT. 

To  determine  the  cost  of  owning  property  in  Duluth  and  vicini- 
ty. 

Let  class  decide  upon  price  of  lot  and  house.  Find  out  different 
kinds  of  taxes  paid  in  Duluth. 

Find  out  how  the  tax  rate  is  determined  and  how  much  it  is. 

Fiml  out  how  it  is  levied. 

What  will  be  the  taxes  on  the  house  aiad  lot  for  one  year  at  that 
rate? 

Compare  tax  rate  with  that  in  Minneapolis. 

Compare  gas  and  lighting  rate? 

In  which  place  can  one  own  property   at  less  cost? 

What  is  the  yearly  interest  on  the  amount  invested  in  the  house 
and  lot  at  6  per  cent? 

Find  out  the  average  rental  price  in  that  i)art  of  the  city  and 
determine  wliether  or  not  it  i)ays  to  own   property. 


sn 

Fiud  the  premium  ou  the  tire  iiisuraiu-e  that  shouM  he  ithiced  ou 
the  property. 

Do  rates  vary  iu  different  i)arts  of  the  city?     Wliy? 

Are  there  other  kinds  of  insurance  than  tire  insurance?  Sliould 
there  be  any  other  insurance  on  the  i)roi)erty':'     If  so  find  tlie  cost. 

If  you  owned  property  at  Pike  Lake  wouhl  you  pay  the  same 
rate  of  taxes? 

Find  the  amount  of  taxes  ou  a  cottaj^e  there,  after  decidinj^  ou 
cost  of  property. 

Determine  also  the  amount  of  insurance  consideriutr  hjcation  and 
value  of  property. 

Suppose  your  father  is  one  of  live  men  who  own  cotta«-es  at  Pike 
Lake.  The  men  decide  to  tax  themselves  8.(i  per  cent  of  the  value  of 
their  cottages,  each  one  to  pay  according  to  the  value  of  tlieir  jtrop- 
erty,  to  hire  a  man  to  protect  them  from  tire  and  thieves.  What  is 
each  man's  tax  if  the  properties  are  valued  at  §2,000,  §2,500,  §1,800, 
§8,200,  and  §2,400  respectively. 

If  you  reckoned  the  interest  ou  the  money  invested  as  rent,  how 
large  a  rent  does  each  man  pay  for  his  lake  cottage  if  the  rate  of  in- 
terest is  6  per  cent. 


MINOR  PROJECTS. 

To  find  how  moving  grains,  lumber,  coal  and  other  commodities, 
are  protected. 

To  find  out  w^hat  the  principal  kinds  of  life  insurance  are. 

To  figure  the  premium  on  a  §500  straight  life  insurance  policy 
for  yourself.  The  teacher  will  give  rates  of  the  different  insurance 
companies. 

To  find  the  interest  on  the  pupil's  Victory  Bond  for  one  year. 
Find  the  interest  for  five  years. 

To  fiud  what  one  would  have  to  pay  for  the  use  of  §20  for  two 
years  at  8  per  cent. 

To  find  what  the  money  the  pupil  now  has  in  the  bank  will  l»e 
worth  in  three  years,  eight  months.  He  has  §85  in  the  bank  and  the 
bank  pays  3  per  cent  interest. 

To  find  the  interest  at  the  date  of  maturity  on  five  War  Savings 
Stamps  bought  iu  January,  1918. 

To  find  how  the  expenses  of  the  national  government  are  met. 

To  find  the  chief  items  of  national  cxjieuse. 

To  grajih  the  military  exi)enses  of  the  Ignited  States  for  tlie  last 
five  years. 

To  compare  resources  of  Minnesota  with  those  of  Wisconsin. 

Standards  of  Attainment. 

The  child  should  have  a  jtractical  working  knowledge  of  the 
principles  of  percentage  as  ai»i)lied  to  commercial  discount,  profit  and 
loss,  commission,  taxes,  duties  or  customs,  insurance  or  interest. 


90 
OBJECTIVE  STANDARDS. 

WOODY  TEST  Series  A 

Additiou 82.4 

Subt  raft  iou 2S.5 

M  a\  t  ij  tlicat  ion 30.6 

Division 27.4 

COURTIS  TEST  Series  B-Speed 

Addition 11 

Sal)t  ractiou 12 

Multiplication 10 

Divisi(:)U 10 

COURTIS  TEST  Series  B-Accuracy 

Addition 75 

Sul)t  raction 8(3 

^lultiitlication 81 

Division 90 

STONE  REASONING  TEST  Actual  Medians  Obtained  . 

r.utte,  1914 5.8 

Bridgeport,  Conn.,  1918 6.8 

Salt  Lake  City,  1915 8.6 

Lead,  S.  Dak.,  1916 9.8 

Tentative  Standard  suggested  by  Stone  (1916)  for  bis  lieasoning 
Tests— 

Tbat  80  per  cent  or  more  of  Seventb  (4rade  pui»ils   reacdi  or 
exceed  7.5  with  at  least  85  per  cent  accuracy. 

Bibliography. 

TEACHER'S  READING.    Brown   and    Coffmau:     How    to    Teach 

Arithmetic. 

Pp.  228-270.  Percentage. 
Klapi)er:  The  Teaching  of  Arithmetic. 

Pp.  829-884.  The  Graph. 
Wilson  and  Wilson:  Motivation  of  School  Work. 

Chap.  IX,  Motivation  of  Arithmetic. 

SUPPLEMENTARY  BOOKS. 

Iloyt  and  Peet:  Everyday  Arithmetic.  Book  TIT. 

T^art  VT.  pp.  85  and  148.  Taxes. 

J*art  V.    pp.  74) 

I*art  VT.  pj).  144)  Insurance. 

Part  VT.  pp.  78,     Duties  and  Customs. 
Stone  and  Millis:  Advanced  Arithmetic.  ,. 

Cliaj).  yi.  Ai)plications  of  Percentage.  |s 

'J'horndike:   Arithmetic.     Book  III.  /> 

Wentworth  and  Smith:   Ailvanced  Arithmetic. 
Wentworth,  Smith  and  Brown:  .luiiior  High  School  Matbemetics. 

Book  1. 


91 
GRADE  VIII-I}. 


DIRECTIONS: 


DIVISION  OF  TIME.  One-fourth  of  the  arithmetic  time  alh.t- 
nieut  of  this  {jrade  sliould  be  devoted  to  oral  work.  Seven-eiyhths  of 
the  time  should  be  f>iveu  to  coux?rete  work. 

REVIEW.  Tables  of  linear,  square  aud  cubic  measure.  Keej)  up 
oral  drill  on  fuudameutal  ])rocesses  in  whole  numbers,  decimals,  and 
fractions.  Use  equations  whenever  they  will  aid  in  ihc  solution  <>f 
problems. 

MAIN  TOPICS.    Involution;  Evolution;  Mensuration. 

INVOLUTION  AND  EVOLUTION.  All  that  is  necessary  to  teach 
under  involution  in  thisyrade  is  the  squares  of  the  numl»ersto  twenty- 
live.  The  children  should  be  taujiht  a  quick  method  of  squarin}»-  any 
number  and  then  should  scpiare  and  memorize  the  numbers  to  twenty- 
live.  Knowing  these  squares  will  save  much  time  in  comi)utation  of 
problems  later  on. 

Square  root  is  needed  in  much  of  the  work  in  mensuration.  Jf  it 
is  applied  to  the  solution  of  actual  problems,  the  cliildren  will  like  it. 
Many  oral  problems  should  be  given,  usino-  the  numbers  to  tweuty- 
Hve,  and  numbers  the  square  root  of  which  may  be  easily  found  by 
inspection.  Square  root  should  be  taught  as  a  process  of  finding  tlie 
side  of  a  square  whose  area  is  knoAvn.  After  that  idea  is  tlioroughly 
fixed  the  mode  of  finding  roots  by  grouping  the  factors  of  the  number 
should  be  taught.  The  meaning  of  jtower,  exponent,  scpiare,  the  si(uare 
root,  and  radical  sign  should  be  taught.  Children  may  be  asked  to 
raise  numbers  to  the  third,  fourth,  or  fifth  powers.  Cubes  to  twelve 
should  be  found  and  learned.  The  pupils  should  work  enough  ex- 
amples to  acquire  facility  in  finding  the  square  root.  The  most  im- 
portant thing  in  this  work  is  that  the  pupils  get  a  clear  concept  of  a 
square.  It  should  also  be  impressed  upon  them  that  only  abstract 
numbers  can  be  squared,  because  a  scpiare  is  the  product  of  two  ecpial 
factors.     The  abstract  number  9  is  a  perfect  Sfjuare.  but   -^9.00  is  not. 

MENSURATION.  The  teacher  should  be  sure  that  she  kn(»ws 
why  mensuration  should  be  taught  before  she  attempts  to  teacli  it. 
^Mensuration  should  be  taught  partly  because  of  its  constructive 
thought  value,  pai'lly  because  of  its  immediate  usefulness;  but  more 
especially  V)ecause  of  its  value  in  intei'preting  the  "natural  features  of 

tthe  world.''  "It  is  not  a  habit  subject  to  be  acquired  through  drill, 
but  a  thought  subject  to  be  developed."— IJrown  aii<l  Coffman.  The 
children  must  thoroughly  understand  all  terms  an<l  measurements 
used.  They  should  be  drilleil  in  estiniating  distances  and  areas  until 
they  can  do  it  with  a  high  degree  of  accuracy.  .Ml  measurements 
shouhl  l)e  accurate  and  all  drawings  shouM  be  ma.Je  to  an  accurate 
scale.  Inaccurate  measurements  an<l  drawings  are  worse  than  none 
at  all.  The  i)rinciples  of  mensuration  should  be  demonstrated  and 
illustrated.  Formulas  should  Ite  developed  by  the  teaclier  witli  the 
pupils.     The  children  should  never  be  asked  to  use  a  formula   winch 


92 

they  have  mechanically  memorized.  The  formula  is  au  outgrowth  of 
the  solution  oi  the  problem.  After  a  i)riiu-ii»le  has  heeu  mastered  hy 
the  pupils,  a  formula  shou  d  he  developed  aud  used.  This  work  is 
«lesiiruetl  to  illustrate  very  clearly  the  value  of  indicatiufj  all  of  the 
steps  iu  a  problem  before  auy  solution  of  it  is  attempted.  Here  also 
is  au  opportunity  for  more  extensive  use  of  the  equation.  In  apply- 
iu}^  the  principles  of  meusuratiou  to.  practical  problems,  only  the 
modes  of  computatiou  of  actual  business  life  should  be  used,  Use  as 
many  short  cuts  as  possible. 

SUBJECT  MATTER. 

AREAS,  I.WOLITIOX,  E\'OLUTIOX. 

To  estimate  the  cost  of  papering,  painting  and  carpeting  a 
five-room  house. 

GENERAL  PROJECT. 

How  many  sq.  y<ls.  of  linoleum  will  it  take  to  cover  the  kitchen 
if  it  is  12  ft.  sq.,  allowing  15  sq.  ft.  for  Hoor  space,  occupied  by  built- 
in  cupboards.  What  will  the  linoleum  cost  if  it  is  2  yds.  wide,  and 
costs  xl.75  per  running;  yard? 

Find  the  cost  of  painting-  the  floor  if  a  pint  of  i)aiut  will  cover 
about  9  sq.  yd.  of  floor  space  in  painting-  the  first  coat  and  about  12 
sq.  yd.  in  painting  the  second  coat,  the  paint  costing  60c  a  quart. 

Find  the  cost  of  painting  the  kitchen  walls  and  ceiling  using 
paint  as  in  the  preceding  problem,  if  the  walls  are  7  ft.  high. 

If  the  floor  area  of  a  square  kitchen  is  576  sq.  ft.  how  long  a  piece 
of  carpet  would  it  take  to  reach  from  one  corner  to  the  corner  diagon- 
ally opposite  ? 

If  the  floor  area  of  a  square  kitchen  is  169  sq.  ft.  how  many  feet 
of  moulding  must  l)e  bought';' 

How  much  would  it  take  if  the  floor  area  is  108  sq.  ft.  and  the 
kitchen  is  12  ft.  long? 

Upon  which  floor  could  you  lay  linoleum  which  is  2  yds.  wude» 
with  greater  economy? 

The  living  room  is  22  ft.  by  IS  ft.  It  has  a  fireplace  across  one 
corner  which  cuts  off  4  ft.  from  the  width  and  5  ft.  from  the  length 
of  the  room.  Find  the  length  of  the  tiled  area  in  front  of  the  fire 
jdace.  Find  the  area  of  the  floor  de«lucting  the  area  cut  off  by  the 
fire  i>lace. 

What  is  the  largest  size  square  rug  which  could  be  used  in  the 
room  if  it  lacked  at  least  1  ft.  of  touching  the  base  board   all  around  ? 

Find  the  area  of  the  room  if  a  book  case  were  put  across  the 
corner  a<ljacent  to  the  fireplace,  cutting  off  the  same  distances  from 
the  width  and  length  as  the  fireplace  does.  Make  a  drawing  of  the 
room  using  the  scale  ■'  in.  to  1  ft. 

Find  the  cost  of  carj^eting  the  room  with  cari>et  27  in.  wide  at 
%1.75  per  yard,  if  the  tiled  area  in  front  of  fireplace  is  18  in.  wide. 

Find  the  cost  of  jjapering  the  room  without  the  book  case,  if  a 
douljle  roll  of  paper  costing  §1.25  is  16  yds.  long  aud  18  in.  wide, 
anrl  the  paper  hanger  charges  >»1.00  per  double  roll.  Allow  for  no 
waste  in  matching  paper. 


98 

Fiud  the  uuinber  of  hoard  feet  of  lunilicr  in  a  square  i)iece  of 
wood  larg-e  enoutfli  to  make  tlie  t(jp  of  a  circular  taldc  :{()  in.  in 
diameter  and  I  of  an  inch  tliick. 

Fiud  tlie  dimensions  of  tlie  top  of  a  square  table  with  the  same 
area  as  the  circular  table  in  the  preeedinfj  problenj. 

The  living  room  has  a  rectanijular  window  4  ft.  ])y  <i  ft.  Fiu<l 
to  three  decimal  ])laces  the  side  of  a  square  window  which  wouM  ad- 
mit the  same  amount  of  light. 

Tn  a  well  lifflited  room  the  window  space  should  be  at  least  ,',  of 
the  floor  space.  How  many  windows  like  that  in  the  i)recedin<,'-  prol)lem 
shouhl  the  living  room  haveV 

Each  i)erson  breathes  on  an  average  of  85  cu.ft.  of  air  in  an  hour. 
If  the  sleeping  room  is  14  ft.  long,  12  ft.  wideand  7  ft.  high,  how  long 
will  the  air  in  it  supply  2  persons. 

The  hallway  is  3  times  as  long  as  it  is  wide  and  has  an  area  of 
108  sq.  ft.     What  are  its  demensions? 

IIow  many  tiles  6  in.  by  4  in.  will  it  take  to  cover  the  hall  flour? 

A  tile  floor  according  to  this  design  is  put  on  the  floor.  Find 
the  area  of  the  white  tile;  of  the  black  tile. 


I 


Z  Z"  X  Z' 

1^  ^^^^^    ^^^^P 


In  the  dining  room  there  is  a  r«ig  12  ft.  by  10  ft.  It  leaves  a 
margin  2  ft.  in  width  on  all  sides.  What  are  the  dimensions  of  the 
floor?     Make  a  drawing  on  the  scale  of  \  in.  to  1  ft. 

What  is  the  area  of  the  floor? 

That  part  of  the  floor  outside  the  rug  and  that  covered  by  it  to  a 
width  of  1  ft.  on  all  sides  is  to  be  varnished.  What  is  the  area  of 
this  portion? 

What  will  the  varnish  cost  if  it  takes  1  gal.  f..r  every  2(MI  s.j.  tt. 
and  the  price  of  varnish  is  *4.7o  per  gal. 

There  is  a  wainscot  in  this  room  which  is   8    ft.    high.     Find    its 


94 

area.     It  the  ruuiu  i;;  7   ft.   ljiy:li,   ^ud   the   remaiuiu<;   portion   of  the 
walls. 

MEASUREMENT. 

Define  terms  of  a  circle;  circumference;  radius;  diameter;  arc; 
sector. 

Find  the  ratio  of  circumference  to  the  diameter  by  measuriujj  the 
circumference  and  diameter  of  cylindrical  bodies,  dividing  and  aver- 
aging quotients.  Call  this  ratio  pi  ("n").  Use  31  as  its  value  in 
rough  calculation,  and  3.1416  for  accurate  calculation.  Teach  the 
l)ui>ils  to  use  judgment  as  to  which  to  use. 

Develop  the  formula  for  finding  the  circumference  when  diameter 
is  given,  to  fiml  the  diameter  or  radius  when  the  circumference  is 
given. 

Develop  the  formula  for  area  of  circle  by  dividing  a  circle  into 
twelve  or  more  equal  sectors  and  fitting  them  together  to  form  a 
rhomboid. 

Projects. 

To  find  the  ratio  of  the  circumference  to  the  diameter  by  divid- 
ing the  circumference  of  a  dollar,  tin  cup,  or  face  of  the  clock,  by  its 
diameter. 

To  average  the  results  obtained  by  the  entire  class  and  compare 
that  average  with  8.1416. 

To  work  out  formulas  for  finding  the  circumference,  diameter, 
radius  and  area  of  the  circle. 

Develop  concrete  problems  from  the  general  project  or  decide 
upon  another  project  to  use. 

BOARD  MEASURE. 

A  continuation  of  the  board  measure  begun  in  Grade  VI- A. 
Develop  formula  for  finding  board  feet. 

Project. 

To  estimate   the  numljer  of  board   feet   of  lumber   required   to 
build  the  framework  of  a  two  room  coom  cottage  at  Sunset  Lake. 
Problems  of  Industry  secured  from  the  Shop  Teachers. 

Standards  of  Attainment. 

Through  the  long  continued  use  of  checks  the  pupil  should  by 
this  time  be  wholly  reliant  ujjon  himself  for  accuracy  and  results. 

The  jtupils  should  have  acciuired  the  habit  of  analyzing  his 
proVjlems  thus:  What  is  known;  what  is  wanted;  what  is  the  best 
method  of  jirricedure. 

He  should  be  able  to  work  quickly,  accurately  and  economically. 


95 
Bibliography. 

TEACHER'S  READING. 

TJrown  and  Coffnian:   How  to  Teach  Aritliiiit'tic. 

Chap.  XIX.  Involution  and  Evolution. 

Chap.  XXI.  Mensuration. 

Chap.  XXIII.  Short  Cuts. 
Klapper:  The  Teaching  of  Arithmetic. 

Pp.  309-820  Measurement. 
Wilson  and  Wilson:  Motivation  of  School  Work. 

Chap.  IX.  Motivation  of  Arithmetic. 

SUPPLEMENTARY  BOOKS: 

Hoyt  and  Peet:  Everday  Arithmetic,  Book  III. 

Chap.  IX  Powers  and  Poots. 
Stone  and  Millis:  Advanced  Arithmetic. 

Pp.  178-l.S(j  Square  Poot. 
Thorndike:  Arithmetic.   Book  III, 

Pp.  19<S-204  Squares  and  Cubes. 
Wenworth  and  Smith:  Essentials  of  Arithmetic. 

Advanced  Book,  Chap.  VII.  Square  Poot  and  Mensuraiotn. 
Wentworth,  Smith  and  Brown:  Junior  IIi<jh  School  ^Mathematics. 


96 
GRADE  VI 1 1- A 


DIRECTIONS. 


DIVISION  OF  TIME.  Oue  third  of  the  Arithinetie  time  allot- 
ment of  this  grade  should  be  devoted  to  oral  work. 

Seven  eighths  of  this  time  should  be  given  to  concrete  problems. 

REX'IEW.  Tables  of  linear,  square  and  cubic  measure;  s<iuares 
to  tweuty-tive,  cuV)es  to  twelve;  square  root,  area  of  triangle;  area,  cir- 
cumference, diameter,  ami  radius  of  a  circle. 

Keei»  u])  <lrill  on  all  fundamental  operations  for  rapidity  and 
accuracy. 

MAIN  TOPIC.    Mensuration  of  Solids. 

MENSURATION. 

Lateral  Surface.  In  teaching  the  lateral  surface  of  prisms,  cylin- 
deas,  pyramids  and  cones,  make  forms  of  stiff  paper  which,  when 
flattened  out,  will  be  easily  recognized  by  the  pupils  as  rectangles, 
triangles,  or  sections  of  a  circle.  They  will  readily  see  how  the 
formulas  i)reviously  learned  can  here  be  applied.  Teach  terms  base, 
slant  height,  lateral  edge  and  lateral  surface. 

Volumes.  Rectangular  solids  must  be  imaged  as  made  up  of 
equal  layers,  composed  of  equal  rows  of  unit  cubes.  The  teacher 
should  build  various  rectangular  solids  of  small  cubes  until  the  child- 
ren see  that  the  volume  of  a  rectangular  solid  is  equal  to  the  pi'oduct 
of  the  number  of  units  of  its  three  dimensions.  Another  way  to  get 
them  to  see  it  is  by  placing  a  number  of  these  cubes  in  the  form  of  a 
rectangle  so  many  cuVjes  long,  so  many  cubes  wide,  and  one  cube  high. 
They  will  easily  see  that  the  number  of  cubes  here  is  identical  with 
the  number  of  squares  in  the  area  of  the  base.  Then  build  on  three 
more  layers  and  the  solid  will  then  contain  three  times  the  number  of 
cubic  units  on  the  base.  Thus  they  have  found  out  that  the  number 
of  square  units  in  the  area  of  the  base  times  the  number  of  linear 
units  in  the  height  equals  the  number  of  cubic  units  in  the  volume  of 
a  rectangular  solid.  From  this  the  formula  for  finding  the  volume 
of  rectangular  solids  and  cylinders  may  be  deduced. 

To  find  the  volume  of  a  pyramid,  compare  it  with  a  prism  of  the 
same  base  and  altitude,  by  filling  the  pyramid  with  sand  and  pouring 
it  into  the  prism.  The  volume  of  a  cone  may  be  compared  with  that 
of  a  cylinder  in  the  same  way.  In  mensuration,  models  should  not 
be  used  any  longer  than  is  absolutely  necessary-  As  soon  as  the 
children  can  projicrly  image  the  figure  under  discussion  the  object 
should  be  dispensed  with.  It  is  not  imjiortant  that  the  children  re- 
member the  formulas  for  finding  areas  and  volumes  of  prisms,  cylin- 
ders, pyramids  and  cones,  but  it  is  important  that  they  learn  to  in- 
terpret and  use  formulas.  Much  may  be  done  with  the  equation  in 
this  work. 


SUBJECT  MATTER. 

J^ateral  area,  voliinie  and  cajiacity  of  in-isms,  cyliniltTs.  iiyraniiils, 
and  coues. 

GENERAL  PROJECT. 

To  estimate  material  used  in  constructing  a  school  building. 

Make  measuremeuts  of  school  lirounds  and  Imildiny-  and  draw  a 
plan  to  scale. 

How  many  loads  of  earth  were  excavated  in  makini;  the  base- 
ment, if  the  loads  averajjed  86  cu.  ft.?  If  this  earth  was  spread  even- 
ly over  the  school  grounds,  how  deep  was  it  covered? 

Find  the  number  of  bricks  required  for  the  walls,  if  it  takes  14 
bricks  per  sq.  ft.     Deduct  for  windows. 

Find  out  as  accurately  as  possible  the  height  and  width  of  wall. 

Find  the  amount  of  cement  used  in  the  foundation. 

How  many  sq.  ft.  of  glass  in  windows  of  the  building? 

Find  the  radiating  surface  of  the  steam  pipes  in  one  room. 

How  much  radiating  surface  is  there  for  every  cu.  ft.  of  space  in 
the  room? 

From  this  result  estimate  the  radiating  surface  required  for  the 
entire  building. 

If  each  pupil  requires  35  cu.  ft.  of  air  per  hour,  how  many  ])U]»ils 
should  be  seated  in  your  room  if  the  air  is  changed  every  houry 

How  much  air  must  be  brought  in  every  minute  to  supply  the 
pupils  in  your  room? 

Find  out  the  following: 

How  many  cu.  ft.  of  space  per  pupil  there  is  in  your  room? 

How  much  paint  was  used  in  painting  the  lower  part  of  the  hall 
walls  if  1  gal.  of  paint  covers  200  sq.  ft.? 

How  many  tons  of  coal  are  used  during  1  year  in  your  school 
building. 

How  large  a  bin  had  to  be  made  to  contain  sufficient  coal  for  the 
entire  year. 

If  the  price  of  coal  next  year  is  8  per  cent  more  than  last  year, 
how  many  more  dollars  will  the  coal  supply  cost? 

Make  out  a  thirty  day  note  for  the  price  of  the  coal,  payable  to 
the  nearest  coal  dealer  at  the  Western  State  Bank.  Discount  it  at  (i 
per  cent. 

How  many  tons  of  hard  coal  can  l>e  put  into  an  inverted  i»yra- 
midal  hopper,  which  is  used  to  feed  a  furnace,  if  the  hopper  is  (3  ft. 
on  a  side,  and  its  altitude  is  12  ft.?  (35  cu.  ft.  of  hard  coal  e<iuals  a 
ton.) 

What  is  the  cost  at  ?!l.l5  per  s(j.  yd.  of  painting  the  pyramidal 
tower  if  the  base  is  a  s<iuare  9  ft.  on  a  side,  and  the  slant  height  is  1.) 
ft.? 

How  many  sq.  ft.  of  plastering  are  there  in  your  room? 

If  your  room  represents  the  average,  how  many  are  there  in  all 
the  rooms  in  your  building. 

How  many  loads  of  gravel  were  used  on  the  grounds  it  1  load 
equals  86  cu.  ft.? 


98 

How  nuK-h  ceineut  was  used  for  the  sidewalks  if  the  cement  was 
H  in.  thick? 

What  did  they  cost  at  (j'2c  per  S(i.  yard? 

How  many  lots  50  ft.  by  150  ft.  could  lie  made  of  the  school 
grounds? 

Find  out  the  average  value  of  lots  in  your  vicinity  and  determine 
the  value  of  the  school  grounds. 

What  tli<l  the  curb  around  tlie  grounds  cost  at  *i7c  a  foot? 

If  the  heating  jtlant  in  your  buildings  is  large  enough  for  20 
rooms  and  there  are  only  14  rooms  in  the  building,  what  percent  of 
its  cai>acity  is  unused? 

What  did  the  i)lant  cost  if  it  was  bought  for  §1*250  with  80  per- 
cent and  10  percent  off? 

At  another  firm  it  could  have  been  bought  for  §1150  with  20  per- 
cent and  5  percent  off.     Which  was  the  lower  price? 

Find  out  the  dimensions  of  the  steam  boiler  and  determine  how 
many  gallon  it  contains? 

Determine  the  number  of  sq.  ft.  in  the  lateral  surface  of  the 
V)oiler? 

BANK  DISCOUNT.  The  children  should  see  the  only  difference  be- 
tween bank  discount  and  simple  interest  is  that  in  one  case  the  inter- 
est is  paid  when  the  money  is  borrowed,  in  the  other  the  interest  is 
paid  when  the  money  is  paid  back. 

Interest  paid  in  advance  on  a  note  is  called  discount.  The  face 
of  the  note  less  the  discount  is  called  the  proceeds. 

Project. 

To  make  out  a  30  day  note  for  S350,  dated  today,  payable  to 
Frank  Lee's  order  at  some  bank.     Discount  it  at  5  percent. 

BUSINESS  ARITHMETIC.  Common businessforms, bills, receipts, 
statements,  bills  of  lading.  Keeping  personal  accounts,  debit,  credit, 
and  balance.  Inventories,  purpose  of  inventories,  when  and  how 
taken.     Sendingmoney.     Investing  money. 

The  work  in  business  arithmetic  should  be  carried  on  largely 
from  the  informational  side.  The  children  should  know  the  forms 
and  jn-actices  in  daily  use.  However,  only  the  simplest  and  most  com- 
mon forms  should  be  considered.  If  possible,  representative  business 
men  should  be  secured  to  talk  to  the  children,  especially  on  banking 
and  investing  money.  The  children  should  be  taught  to  beware  of 
"(Tct-rich-quick'"  jjropositions.  The  element  of  safety  is  the  first  con- 
sideration of  any  investment.  The  security  of  a  loan,  for  instance,  is 
to  be  considered  before  the  rate  of  interest.  It  is  always  better  to 
take  a  lower  rate  of  interest  than  to  take  a  risk.  In  general  the  higher 
the  rate  of  interest  the  greater  risk.  Security  is  required  by  those 
loaning  money.  Teach  the  meaning  of  the  terms  real,  collateral,  and 
l>ersonal. 

A  few  realistic  problems  should  be  solved  to  illustrate  the  differ- 
ent topics  as  they  are  taken  up. 


99 
MINOR  PROJECTS. 

Usiug  the  foUowiiiff  sample  price  list,  estimate  the  cost  of  supplies 
in  j'our  room  for  one  year. 

Composition  books *4.S0  jjer  ^ross  less  5  percent. 

Drawing  paper 1.80  per  doz.  pkgs.  less  10  i)er  cent. 

Drawinfj  pencils 4.80  per  gross  less  '20  per  cent 

Penholder 8.40  per  gross  less  12  per  cent. 

Pens 6.TC  per  gross  less  2.5  per  cent. 

Rulers 40c  per  doz.  net. 

Estimate  the  cost  in  the  entire  building  for  one  year. 

To  write  a  receipt  for  Junior  lied  Cross  dues. 

To  keep  personal  accounts. 

To  make  an  inventory  of  contents  of  the  room. 

To  write  a  check  to  pay  for  a  lost  book. 

To  make  out  a  promissory  note  payable  to  the  Athletic  Associa- 
tion or  Manual  Training  de])artment. 

To  graph  certain  stock  quotations  for  a  week. 

To  find  out  what  causes  the  fluctuations  which  your  graph  shows. 

To  find  how"  much  it  would  cost  to  insure  a  bicycle  for  a  year. 

To  borrow  money  from  the  school  bank,  give  a  note  and  find  the 
proceeds. 

OPTIONAL  WORK. 

Solution  of  algebraic  problems  of  one  unknown  quantity. 
Drill  for  a  higher  rate  of  si)eed  and  accuracy. 

Standards  of  Attainment. 

The  pupils  should  have  the  habit  of  checking  by  some  certain 
means  fully  established. 

They  should  use  common  sense  rather  than  recall  some  mean- 
ingless rule  or  formula. 

They  should  have  the  ability  to  interpret  and  use  formulas. 

They  should  be  able  to  use  the  laws  of  equations  in  solving 
problems  according  to  a  good  form  developed  from  accurate  thinking. 

The  habit  of  looking  for  short  cuts  should  be  well  established. 

The  pupils  should  by  this  time  be  wholly  reliant  upon  themselves 
for  accuracy  of  results. 

OBJECTIVE  STANDARDS. 

WOODY  TEST.  Series  A 

Addition 33.9 

Subtraction 31.7 

Multiplication 32.9 

Division 30.9 

COURTIS  TEST:  Series  B  -Speed 

Addition 1- 

Subtraction 13 

Multiplication H 

Division H 


100 

COURTIS  TEST  Series  B— Accuracy. 

A<l<litiun Tt). 

Sulitractiou !^7. 

Multiplication <"^1. 

Division 91. 

STONE  REASONING  TESTS  Actual  Medians  Obtained 

r.utte,  1914 7.7 

r>ri(lg-eport,  Couu.,  1918 4.5 

Salt  Lake  City,  1915 10.6 

Lead,  S.  Dak.,  1916 11.4 

Nassau  Co.,  N.  \\,  1918 8.2 

Tentative  Stamlards  sug-ffeste<l  hy  Stone  (1916)  for  his  reasoning 
tests:  That  ^0  per  eent  or  more  of  the  Eijj-hth  grade  pupils  reach  or 
exceed  a  score  of  8.75  with  at  least  90  per  cent  accuracy. 

Bibliography. 

METHOD.    TEACHER'S  READING. 

Brown  and  Coffman;  How  to  Teach  Arithmetic 

Chap.  XXI  Mensuration 

Chap.  XIX  Involution  and  Evolution 

Chap.  XVII  Banking 

Chap.  XXIII  Short  Cuts 

Chap.  IX  Waste  in  Arithmetic 
Judd:  Socializing  Arithmetic 
Klapi)er:  The  Teaching  of  Arithmetic 

pp.  299-820  Mensuration 

pp.  821-826  Business  Forms 

l»p.  886-877  Need  of  Scientific  Standards 
Wilson  and  Wilson:  Motivation  of  School  Work 

Chap.  IX  Motivation  of  Arithmetic 

SUPPLEMENTARY  BOOKS. 

Alexander  and  Dewey:  Arithmetic 

Durrell:  Arithmetic,  Book  III 

Harvey:  Practical  Arithmetic 

Iloyt  and  Beet:  Everyday  Arithmetic,  Books  Two  and  Three 

Iloyt  and  Peet:  Business  Arithmetic 

Thorndike:  Arithmetic,  Book  III 

Vosburgh  and  (rentlemen:  Junior  High  School  Mathematics: 

Business  Arithmetic 
Weutworth,  Smith  &  Brown:  Junior  High  School  Mathematics. 


DIRECTIONS. 


Kil 
(JKADK   IX-i;. 


I 


DIVISION  OF  TIME.  Oiu'-tliinl  of  tlu'  time  alL.tiiiciit  f-.r  aritli- 
nietic  ill  this  yrade  sliouM  l)e  uiveii  to  intensive  <liill  woik  in  lapiM 
eak'ulatiou. 

Mueli  of  the  remainder  of  tlie  time  .should  l»e  t,'-iven  to  .solvin<r 
practical   problems  without  a  pencil. 

MAIN  TOPICS.  Additiou,  subtraction,  multiplication,  ami  ili\  i- 
sion  of  whole  numbers,  fractions  and  decimals. 

Percentage. 

FUNDAMENTAL  OPERATIONS  IN  WHOLE  NUMBERS. 

This  course  is  designed  primarily  for  pupils  takiuff  the  commer- 
cial courses,  who,  therefore,  need  to  be  able  to  solve  accurately  and 
quickly  the  problems  arising  in  ordinary  business  transactions.  The 
work  is  not  to  be  considered  in  the  nature  of  a  review  of  the  work  (»f 
the  grades.  It  should  be  taken  up  and  carried  on  as  an  advanced 
subject.  Something  of  the  history  of  the  origin  and  development  of 
numbers  and  the  processes  used  in  arithmetic  should  be  given.  Un- 
usual care  should  be  taken  to  develop  skill  in  the  use  of  the  tools  of 
arithmetic.  Special  attention  is  called  to  the  constant  appeal  to 
the  pupils  to  become  independent  of  the  pencil  in  solving 
problems,  hence  due  amount  of  practice  should  be  given  in  oral  work. 
Short  methods  which  are  practical  should  be  used.  One  good  short 
method  should  be  found  and  that  only  should  be  taught.  Since  abso- 
lute accuracy  is  a  prime  essential  in  business  transactions  verification 
and  checking  should  be  practiced  whenever  jK)Ssible. 

The  classes  in  commercial  arithmetic  should  be  tested  at  the  Ije- 
ginniug  of  the  year,  so  that  the  needs  of  each  individual  may  be 
known  to  himself  and  to  the  teacher  at  once.  The  pupils  should  then 
be  grouped  and  work  assigned  to  suit  the  needs  of  each  group.  There 
is  always  work  for  those  who  excel  as  well  as  for  those  who  <lo  not. 

Waste  no  time.  Go  to  work  the  first  day.  Plan  each  day's  work 
so  that  the  pui»ils  will  feel  when  the  lesson  is  over  that  it  has  been 
worth  while.  Because  this  work  is  mechanical  is  no  excuse  for  its 
l)eing  dull  and  uninteresting.  Even  the  oMer  i)uitils  enjoy  the  game 
an<l  play  idea.  The  commercial  arithmetic  teacher  must  keeji  ahead 
of  her  pupils  in  skill  and  accuracy. 

ADDITION.  Master  the  forty-five  combinations.  Find  groui»s  of 
figures  aggregating  ten  and  twenty  in  the  body  of  a  column  as  in  the 
following: 

See  Moore  and  ^Miner  fori»racticei>roblenis 
of  this  kind.  The  children  should  be  taught 
to  look  for  easy  ways  of  doing  things.  The 
work  should  be  varied  to  keej)  up  interest. 

In    all    written    work  make  jilain,  legible 

figures    of    a    uniform    size,    write  them  equal 

distances  from  each  other,  ami  be  sure  that  the 

units  of  the  same  onler  stand  in  the  same  ver- 

6  tical  column. 


4 
•> 

2 

17 

o 
8 

2 

4 

9 

8 

7 

9 

8 

2 

1 

5 

102 

"In  Imsiuess  it  is  imi)ortant  that  tig-ures  be  made  rapidly; 
l»ut  raj»i(lity  should  never  be  secure<l  at  the  exi)eiise  of  leg:ibility." 
---More  aud  3Jiner. 

Casting-  out  9's  is  a  coniinou  cheek  tor  addition.  But  the  sim- 
plest way  of  checking-  a<Mition  is  to  add  the  columns  in  reverse  order. 
If  the  results  obtained  by  both  processes  agree,  the  work  may  be  as- 
sumeil  to  l>e  correct.     The  following  is  an  illustration  of  this  method: 

34  54966  40 

40  78728  18 

87  47929  37 

18  78425  40 

40  45623  34 

383920  78249  383920 

Sl'BTRACTIOX.  Use  the  "making  change"  method  for  the  sake 
of  uniformity.  If  a  pui)il  uses  the  "take  away"  method  skillfully  do 
not  require  him  to  chang-e. 

MULTIPLICATION'.  Multiplication  should  be  presented  as  a 
short  method  of  aildition.  Use  some  practical  short  methods  and  ali- 
quots.  Learn  some  good  method  of  checking  the  work.  Casting-  out 
9's  is  sugg-ested. 

DIVISION.    Teach  the  meaning-  of  division,  as: 

X24.00  divided  by  'S2.00  equals  12,  the  number  of  times  $2.00  is 
contained  into  §24.00, 

824.00  divided  by  2  equals  812.00.  If  824.00  is  divided  into  two 
jiarts.  each  jiart  contains  812.00. 

The  work  in  the  four  fundamental  operations  given  in  Moore 
and  Miner's  Practical  Business  Arithmetic  is  good  and  may  well  l)e 
followed. 

FRACTIONS  AND  DECIMALS.  Combine  fractions  aud  decimals 
and  show  that  decimals  is  but  another  way  of  writing  fractious. 

In  addition  and  suVjtractiou  of  fractious  emphasize  that  only  like 
numbers  are  added  and  subtracted.  Special  attention  should  be  given 
to  cancellation. 

In  all  the  operations  iu  decimals  make  sure  that  the  pupils  know 
exactly  where  to  place  the  decimal  point.  Fractions  with  impossible 
denominators  and  com]»lex  fractions  should  be  omitted.  Only  such 
fracti(tns  and  decimals  as  are  actually  used  in  business  transactions 
shduM  be  used. 

PERCENTAGE.  Teach  the  subject  as  outlined  in  ]Moore  and 
Miner's  Practical  Business  Arithmetic.  Be  sure  that  the  pupil  under- 
stands that  a  jjercent  of  a  uumljer  is  so  many  hundreilths  of  it.  To 
illu.strate.  6  i>er  cent  eiiuals  iJio,  or.  ,06,  wiitteu  decimally.  Under 
percentage  take  uj)  loss  and  gain,  marking  goods,  commercial  discount 
and  interest.  In  commercial  discount  use  the  sum  an<l  difference 
methoil  for  oral  prolilems  and  cancellation  for  written  problems.  In 
interest  use  the  60  <lay  method  and  develop  by  the  cancellation 
method. 


103 

lu  connection  with  the  work  in  i)ercentatre  use  {frai»hin<r.  Tlie 
pupils  shouhl  make  <irai)hs  until  they  are  aide  to  understand  an<l  np- 
]»reciate  the  graphs  so  commonly  used  now  in  niatfazines  and  the 
daily  papers. 

SUBJECT  MATTER. 

ADDITION'.  l{ai)id  adilition  depends  entirely  ujton  the  aliility  to 
group.  The  same  principle  which  is  employed  in  reading  a  sentence 
applies  to  aihling  a  single  cohunn.  ''From  two  to  four  figures  should 
be  read  at  sight. as  a  single  number,  and  the  group  so  t'orme«l  should 
be  rapidly  combined  with  other  groups  until  the  result  of  any  given 
column  is  determined."  —  Moore  and  Miner. 

Only  correct  results  can  be  accepted,  of  course.  Accuracy  must 
never  be  sacrificed  for  rapidity.  Add  horizontally  as  well  as  verti- 
cally. 

PROJECTS.  To  find  how  I  rank  in  addition  if  the  following 
groups  should  be  read  at  the  rate  of  150  per  minute. 

1   1   2  2  4  1   3  3  4  3   1   4  2  4  7  S  9  S  .5  G  4  5  5  7 
1  3  1   2  1   5  2  3  2  6  7  3  5  6  7  9  9  <S  5   1  4  3  4  2 


1 

5 

6 

6 

8 

9 

8 

7 

7 

4 

9 

( 

6 

7 

5 

3 

o 

4 

5 

7 

0 

8 

6 

(5 

9 

6 

1 

2 

3 

5 

8 

3 

8 

7 

9 

9 

8 

9 

9 

8 

4 

2 

All  the  possil)le  groujis  of  three  figures  each  are  given  on  pages 
14  and  15  of  Moore  and  Miner. 

To  determine  my  ability  to  add  if  the  sums  of  the  grouits  of 
three  figures  each  should  be  name<l  at  the  rate  of  120  per  minute. 

To  graph  my  ability  to  add  as  compared  with  what  my  ability 
should  be. 

To  determine  who  in  our  group  can  first  get  to  the  to]*  <>f  a 
column  of  figures  with  the  correct  result. 

To  see  how  fast  I  can  name  the  results  in  the  following  l)y  taking 
advantage  of  the  lO's  and  20's  wherever  possible: 

1  2  7  ()  7  5  2  3  2  4  1  9  1  2  3 
9  8  3  4  3  4  8  5  7  3  1  2  (J  2  S 
7  4  5  2  9  9  7  5  3  3  S  9  3  9  9 
3  ()  5  8  1  1  3  4  5  7  1  4  (5  9  7 

2  7  7  3  8  tj  4  6  5^2  (j  1  2 

2 1  3  5  2  

To  see  which  group  can  obtain  the  highest  per  cent  f«>r  a-lding  a 
certain  number  of  groups  of  a  given  length   in  a  given  time 

To  see  how  well  and  how  rapi<lly  I  can  write  the  t<. 11. .wing  from 
rapid  dictation: 

?!75.18.  .S123.95.  !sl47.25.  S9.50.  ?;1S1.5(),  .sl72.2»i,  x,s.-).!»s.  .s:{24.!t7 
?i48.lU,     x(j9.20,  x312.«i0,  x415.90. 


9  3S 

25 

4     S 

4 

7     ] 

1 

U      1 

3 

104 

To  prove  that  I  have  writteu  tlie  sums  c-orrectly  l)y  addiiiy  aud 
compariufjf  with  the  correct   result: 

To  <leterraiue  by  casting  out  O's  whether  tlie  result  iu  the 
t'ollowius:  exaini»le  is  correct: 

54,(j(jy 
15,218 
36,425 
45,325 
6S,619 


220,256 
SUBTRACTION. 

Projects: 

To  tiud  how  much  change  I  should  receive  from  §1.00.  if  I  spent 
§.74. 

To  tind  the  coins  which  I  must  give  in  exchange  for  a  S2.00  bill, 
if  some  one  has  jiurchased  §1.19  worth  of  flowers. 

To  find  how  much  money  I    have  left  iu  the  school  bank  if  I  had 
§200.00  on  deposit  and  draw  §157.00. 

To  see  how  quickly  I    can   find   the   missing  numbers  in  the  fol- 
lowing: 

50     50     50     50     50     50     50     50     50 
__     47     __     34     24     __     __     19     __ 


23     __     24     __     __       7       8     __     21 

FRACTIONS,  DECIMALS,  PERCENTAGE  AND  INTEREST.  What 
has  been  said  in  regard  to  whole  numbers  is  also  apjilicable  to  frac- 
tions, decimals,  percentage,  and  interest.  Before  taking  uj)  the  study 
of  percentage,  look  this  subject  up  carefully  from  the  outlines  of  the 
grades  below.  In  connection  with  the  work  in  percentage  continue 
the  use  of  the  graph.  Moore  and  Miner  has  some  good  suggestions 
for  this  work.     Use  material  from  magazines  and  daily  pai)ers. 

In  all  of  this  work  use  the  methods  taught  in  lower  grades  wher- 
ever at  all  practical. 


105 

GRADE  IX- A. 
DIRECTIONS. 

DIVISION  OF  TIME.  At  least  fifteen  minutes  n(  the  allotiiieiit  of 
aritlimetic  time  in  tliis  grade  should  he  yiveu  to  rapid  ealeulation. 

The  remainder  of  the  time  should  be  given  to  the  application  of 
the  knowledge  gained  during  the  iirst  semester  and  to  store  the  impil's 
mind  with  business  information. 

MULTIPLICATION  AND  DIVISION.  Moore  and  .Miner's  Pract- 
ical Business  Arithmetic  is  full  of  suggestive  material  for  interesting 
projects.  Projects  similar  to  those  suggested  in  a<lditi()ii  ami  subtract- 
ion should  l)e  given. 

All  of  tliis  work  should  be  put  before  the  pui)il  in  the  form  of 
interesting  projects  in  order  to  keep  him  interested  an<l  enthusiastic. 

MAIN  TOPICS.  Banking,  insurance,  commission,  customs,  duties 
and  taxes,  stocks  and  bonds,  and  business  forms. 

BANKING.  Teach  bank  discount.  Partial  payments  should  be 
computed  according  to  U.S.  rule,  since  most  installment  plans  are  based 
on  this  method  of  payment.  Teach  the  methods  of  balancing  a 
V)auk  pass  book  and  making  the  banker's  daily  balance. 

Teach  the  following  methods  of  sending  money:  the  l)ank  di-aft, 
I)OStal  money  order,  and  the  express  money  order.  Discuss  savings 
bank  accounts,  and  compound  interest  as  ai)plied  to  savings  accounts, 
also  clearing  house  problems.  Show  how  building  and  loan  associa- 
tions differ  from  banks.  Give  a  few  problems  to  be  solved  under 
each  topic  discussed.     The  problems  in  Moore  and  .Aliner  are  good. 

INSURANCE.  P^xplain  the  following  kinds  of  policies:  ordinary 
life,  twenty-payment  life,  and  endowment. 

Careful  attentioJi  should  be  given  to  questions  of  extension  of 
time,  cash  surender  values,  and  dividends  which  ai'e  of  imixirtance  to 
every  policy  holder. 

Local  insurance  agents  will  upon  application,  sui)iily  sami)le  pol- 
icies and  rate  books  free  of  cost. 

This  adds  much  interest  to  the  work  and  sliouM  be  used.  It  is 
the  l)est  kind  of  material  in  teaching  insurance. 

COMMISSION,  CUSTOMS  AND  DUTIES,  AND  TANES.  Discuss 
commission,  customs  and  duties,  and  taxes.  See  outlines  for  the  Sev- 
enth Grade  for  suggestions  for  this  work.  Under  ta.xes  take  up  tlie 
Federal  Income  Tax.  Work  enough  practical  problems  to  make  clear 
the  information  given  upon  each  tojiic. 

STOCKS  AND  BONDS.  Distinguish  bt-twccn  tlu-in.  K.xplain 
common  and  preferred  stock.  Discuss  reason  why  stocks  tluctuatc  in 
l»rice,  and  the  meaning  of  assessments  and  dividends.  The  prolileiiis 
from  the  text  should  be  supiilemented  by  i)robleiiis  i.repared  by  \isiiig 
stock  ([notations  from  the  daily  papers. 

Teach  bonds  simply  as  notes  issued  by  cori»oratioiis.  munciiiali- 
ties,  states,  and  nations.  Uomiiare  a  bond  with  a  promissory  note  and 
point  out  the  similarities.  Discuss  registere.l  bonds  and  Liberty 
Bonds.      Study  Ijond  (juotationsin  the  pai>ers  and  figure  the  ratesof  in- 


1U(3 

terest  they  payou  iuvestiueutsat  various  quotations.  Interest  the  pupils 
in  hrino-iuof  in  all  of  the  outside  information  they  can  obtain.  Encoui-agre 
tliem  t*t  lie  t)n  the  look  out  for  valuable  information  along  commercial 
lines.  Teach  them  to  be  able  to  adapt  themselves  to  new  conditions. 
BUSINESS  FORMS.  Blank  forms  of  checks,  notes,  commercial 
and  bank  drafts,  and  receipts  should  be  prepared  in  the  priutiugdei)art- 
ment  and  given  pupils  for  practice  in  filling  them  out  under  the  di- 
rection of  the  teacher.  Blank,  full,  special  and  restrictive  indorse- 
ments should  be  made  familiar  to  the  pupils  through  practice  in  in- 
dorsing. Use  available  pay-roll  forms  from  business  houses.  Spend 
some  time  in  figuring  payrolls,  billing,  and  making  out  cost  sheets. 

OPTIONAL  WORK. 

Use  more  difficult  jiroblems  pertaining  to  the  different  topics. 

Do  some  work  in  accounting  to  emphasize  form  and  neatness.. 

Make  graphs  representing  war  expenditures,  increase  or  decrease 
in  i>rice  of  certain  commodities  during  a  certain  period,  or  fluctuation 
of  stock.     These  are  interesting  and  instructive. 

Standards  of  Attainment. 

The  pupil  must  have  acquired  a  high  degree  of  accuracy  and 
speed  in  haiulling  the  fundamental  process;  through  checking  his  own 
work  he  must  have  become  self-reliant;  through  the  study  of  the 
practical  and  everyday  features  of  arithmetic  he  must  have  a  working 
knowledge  of  the  usages  and  the  phraseology  of  business  and  com- 
merce. 

See  Objective  Standards  for  (xrade  VIII. 

Bibliography. 

Moore  and  Miner:  Practical  Business  Arithmetic. 

Power  and  Locker:  Practical    Exercises    in    Rapid    Calculation, 

(1917). 
Mcintosh:  Exercises  in  Papid  Calculation. 
See  bil)liography  for  Seventh  and  Eighth  Grades. 


107 

GRADES  IX-i;  aii.l  A. 

The  foUowiny  is  a  statenieut  of  the  subject  matter  li>  Vm-  covered 
by  the  Ninth  Grade  in  Algebra: 

TEXT 

Ilawkes,  Luby  and  Touton:  First  Course  in  Altjfebra, 
Nine-B:  pages  1  to  121. 
Nine- A:  pages  122  to  289. 

At  least  six  weeks  of  the  Niue-1>  (irade  should  be  given  to 
factoring.  The  success  of  the  Niiie-A  Grade  work  depends  hirgely 
upon  the  pupils'  ability  to  factor. 

It  is  not  expected  that  all  of  the  exercises  under  eacli  to])ic  shall 
be  solved.  It  is  better  to  solve  a  few  and  check  them  than  to  solve 
many  without  checking.  Long  involved  solutions  and  i»r(d)lems 
which  are  of  no  vital  interest  to  the  pujjils  should  be  omitted. 
Teachers  may  substitute  other  problems  within  the  comprehension 
and  experience  of  the  children.  Dividing  and  multii)lying  by  ex- 
pressions of  more  than  two  terms  is  not  practical. 

The  work  of  the  entire  course  should  center  about  the  equation, 
taking  up  other  topics  only  as  there  is  need  for  them  in  the  solution 
of  equations. 

There  is  no  need  for  many  formal  definitions.  Most  of  the 
terms  in  algebra  may  be  learned  from  intelligent  use.  A  <lefinition 
should  be  taught  only  when  it  is  necessary  in  order  to  get  a  clear 
concept. 


lOS 

General  Bibliography: 

METHOD.  Browu  aud  Cotfmiiu:  How  to  Teach  Arithmetic:  Row, 
Peterson  and  Co. 

Courtis,  S.  A.:  Measurement  of  Efficiency  and  (Growth  in  Arith- 
metic: The  Elementary  Scliool  Teacher,  Chicago;  Vol.  X, 
1909,  pp.  55-74,  pp.  177-199;  Vol.  XI  1910,  pp.  171-185. 
pp.  8U0-370,  pp.  5'2S-539;  Vol.  XII,  1911,  pp.  127-137. 

Freeman,  F.  X'.:  The  Psychology  of  the  Common  IJrauches: 
IIoughton-Mifflin  Co. 

Jessup,  W.  A.:  Economy  of  Time  in  Arithmetic:  tState  I^niversity 
of  Iowa,  Iowa  City,  Iowa. 

Jessup  aud  Coffman:  The  Supervision  of  Arithmetic:  The  Mac- 
millan  company. 

Kendall  and  Mirick:  How  to  Teach  the  Fundamental  Subjects: 
Chap.  Ill  Mathematics:  Houghton-Mifflin  Co. 

Klapper:  The  Teaching  of  Arithmetic:  D.  Appletou  and  Co. 

McMurry:  Special  Method  in  Arithmetic:  The  Macmillan  Co. 

Smith,  David  E.:  The  Teaching  of  Arithmetic:  Ginn  and  Co. 

Stone,  C.  W.:  Arithmetical  Abilities  aud  Some  Factors  which 
Determine  Them:  Teachers  College  Bureau  of  Publication. 
X.  V. 

Suzzalo,  Henry:  The  Teaching  of  Primary  Arithmetic:  Houghton 
Mifflin  Co. 

U.  S.  Bureau  of  Education:  A  Course  of  Study  in  .\rithmetic:  Bos- 
ton, Mass. 

Wilson,  G.  M.:  A  Course  of  Study  in  Elementary  Mathmetics, 
Connersville,  Indiana  Schools. 

Wilson  and  Wilson:  The  Motivation  of  School  Work;  Chap.  IX: 
Houghton  Miffliu  Co. 

DISCUSSIONS  OX  THE  USE  OF  STANDARD  ARITHMETIC  TESTS. 

Ashbaugh,  E.  J.:  The  Arithmetical  Skill  of  Iowa  School  Child- 
ren; Bulletin  X'o.  24,  Extension  Division,  University  of 
Iowa. 

Ballon,  Frank  W.:  Educational  Standards  and  Educational 
Measurements,  with  Particular  Reference  to  Standards  in 
the  Four  Fundamentals  in  Arithmetic;  Bulletin  10,  1914: 
Boston  Public  Schools  Bulletins  3  and  7,  Department  of 
Educational  Investigation  and  Measurement,   Boston. 

Buckingham,  B.  R.:  Xotes  on  the  derivation  of  Scales,  with 
special  reference  to  Arithmetic:  Fifteenth  Yearbook  of  the 
National  Society  for  the  Study  of  Education,  Part  1. 

Counts,  (4.  S.:  Arithmetic  Tests  and  Studies  in  the  Psychology 
of  Arithmetic;  University  of  Chicago  Supplemeutaj-y  Edu- 
cational Monographs,  Vol.  I,  No.  4_ 


1 


109 

Courtis,  S.  A.:  Third,  Fourth  aud  Fifth  Annual  Accountiuffs, 
1918-191G:  Dei)artnK'ut  of  Co-operative  Research.  .S2  Eliot 
Street,  Detroit.  The  llelial)ility  of  Siiiyle  Measurements 
with  Standard  Tests:  Klenientary  School  .Journal,  March  ami 
June,  1918. 

Haffgerty,  M.  K.:  Arithmetic:    A    Co-operative   Study  in    Educa- 
tional Measurements:   Indiana  Cniversity  Studies,  No.  27. 
Studies  in  Arithmetic:   Indiana    Cniversity   Studies,   No.  H'l, 
Vol.  8.  Sept.,  1916. 

Judd,  Chas.  IL:  Measuring  the  Work  of  tiie  Puhlic  Schools: 
Cleveland  Foundation,  Survey  Ke])ort,  Clevehunl,  Ohio. 

Monroe,  W.  S.:  Educational  Tests  and  Measurements,  Chaj).  II, 
Arithmetic:  lIoughton-MitHin  Co. 

Report  of  the  Use  of  the  Courtis  Standard  liesearch  Tests: 
Kansas  State  Normal  School,  Emporia;  Bulletin,  new  series, 
Vol.  4,  No.  8. 

Smith,  J.  11.:  Individual  Variations  in  Arithmetic:  Elementary 
School  Journal,  Vol.  17,  pj).  195-'2(J0. 

Stone,  C.  W.:  Arithmetical  Abilities  and  Some  Factors  Deter- 
mining Them:  Teachers'  College  Contributions  to  Educa- 
tion, No.  19. 

Standardized  Ueasoning  Tests  in  Arithmetic  and  How  to  Utilize 
Them:  Teachers' College  Contributions  to  Education, No.  S^: 
Teachers'  College,  Columbia  University,  N.  Y. 

Theisen,  W.  W.:  The  Use  of  Some  Standard  Tests;  Chap.  II. 
Arithmetic:  State  Department  of  Public  Instruction,  Madi- 
son. Wisconsin. 

AVoody,  C:  Measurements  of  Some  Achievements  in  Arithmetic: 
Bureau  of  Publications,  Teachers'  College. 

OTHER  DISCUSSIONS  OF  ARITHMETIC.  School  Survey  Reports: 
Butte,  Cleveland,  Denver,  (irand  Rapids,  Janesville,  Salt 
Lake  City,  St.  Paul,  St.  Louis,  Nassau  Co.,  N.  V. 

Second  Report  of  the  Committee  on  Elimination  of  Subject  .Mat- 
ter: Chap.  V  Arithmetic:  Iowa  State  Teachers'  Asst)ciation. 

Bush,  M.  G.:  The  Fundamental  Number  Facts:  School  and  So- 
ciety, Sept.  1,  1917. 

Chase,  S.  E.:  Waste  in  Arithmetic:  ''I'eaclK'rs'  College  Record, 
Sept.,  1917. 

Oist,  S.  A.:  Errors  in  the  Fundamentals  of  Arithmetic:  School 
and  Society,  August  11,  1917. 

Holloway,  II.  U.:  The  Relative  DitHculty  of  the  KlciiKMitary 
Processes  State  (Gazette  Pub.  Co.,  Trenton.  N.  ^. 

Kirby,  T.  J.:  Practice  in  the  Case  of  School  Children:  Rureau  of 
Publications,  Teachers'  College,  N.  V. 


110 

standard  Tests. 

FUNDAMENTAL  OPERATIONS. 

Courtis  Standard  Research  Tests,  Series  I>.  The  tests  may  be 
secured  from  S.  A,  Courtis,  82  Eliot  Street,  Detroit  Micliigau. 

Research  Tests  in  Arithmetic,  Addition  of  fractions,  desiy^ned  hy 
F.  W.  IJalloii.     ("oities  of  these  tests  are  not  obtainable. 

The  Cleveland  Survey  Arithmetic  Tests.  Copies  of  the  test 
paj^ers  may  be  obtained  from  Charles  II,  Judd,  School  of 
Education,  University  of  Chicago,  Chicago,  Illinois. 

Stone's  Arithmetic  Test  for  fundamental  Operations.  Designed 
as  a  general  test.  Copies  may  be  obtained  from  the  Rureau 
of  Publications,  Teachers  College,  Columbia  University. 
New  York  City. 

Arithmetic  Scales  devised  by  Clifford  Woody.  Copies  may  be 
obtained  from  the  Bureau  o:^  Publications,  Teachers  College, 
Columbia  University,  New  York  City. 

REASONING  TESTS. 

Stone's  Reasoning  Test.  For  copies  of  the  test,  address  Bureau 
of  Publicatious,Teachers  College,  Columbia  University,  New 
York  City. 

Starch's  Arithmetical  Scale  A.  Copies  may  be  obtained  from 
Daniel  Starch,  University  of  AVisconsiu,  Madison, ^Visconsin. 

Buckingham's  Reasoning  Test  i  n  Arithmetic.  Used  by  Buck- 
ingham in  the  Survey  of  t  he  Gary,  and  the  Prevocational 
Schools  of  New  York  City. 


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